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LIBRARY OF CONGRESS. 

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UNITED STATES OF AMERICA. 



ELEMENTS OF GEOMETRY 



TRIGONOMETRY. 



APPLICATIONS IN MENSURATION 



BY CHARLES DA VIES. LL. D. 

(lOTnOR OF FIRST LESSONS IN ARITHMETIC, ELEMENTARY ALGEBRA, 
PRACTICAL MATHEMATICS FOR PRACTICAL MEN, ELEMENTS OF 
SURVEYING, ELEMENTS OF DESCRIPTIVE GEOMETRY, 
SHADES, SHADOWS, AND PERSPECTIVE, ANA- 
LYTICAL GEOMKTRY, DIFFERENTIAL 
AND INTEGRAL CALCULUS. 



<V 




S. BARNES & COMPANY, 
NEW YORK AND CHICAGO. 



DAYIES' MATHEMATICS. 



1KB WSil F@IM ©©WSSB 8 

And Only Thorough and Complete Mathematical Series. 



IIST THREE PASTS. 



/. COMMON SCHOOL COURSE 

Davics' Primary Arithmetic. -The fundamental principles displayed in 

the Object Lessons. 
Davies' Intellectual Arithm etic — Referring all operations to the unit I as 

the only tangible basis for logical development. 
Davies' Elements of Written Arithmetic— — A practical introduction 

to the whole subject. Theory subordinated to Practice. 
Davies' Practical Arithmetic* — The most successful combination cf 

Theory and Practice, clear, exact, brief, and comprehensive. 

//. ACADEMIC COURSE. 

Davies' University A rithmetic.-— Treating the subject exhaustively as 

a science, in a logical series of connected propositions. 
Davies' Elementary Algebra.*— & connecting link, conducting the pupil 

easily from arithmetical processes to abstract analysis. 
Davies' University Alf/ebra. — For institutions desiring a more complete 

but not the fullest course in pure Algebra. 
Davies' Practical Ma them a tics. —The science practically applied to the 

useful arts, as Drawing, Architecture, Surveying, Mechanics, etc. 
Davies' Elementary Geometry. — The important principles in simple form, 

but with all the exactness of vigorous reasoning. 
Davies' Elements of Surveying.— Re-written in 1870. The simplest and 

most practical presentation for youths of 12 to 16. 

///. COLLEGIATE COURSE. 

Davies' Bourdon's Algebra.* — Embracing Sturm's Theorem, and a most 
exhaustive aud scholarly course. 

Davie*' University Algebra.* — A shorter course than Bourdon, for Insti- 
tutions have less time to give the subject. 

Davies' leyendre's Geo met ry.— Acknowledged the enly satisfactory trea- 
tise of its grade. 300,000 copies have been sold. 

Davies' Analytical Geometry aud Calculus. — The shorter treatises, 
comoined iu one volume, arc more available for American courses of study. 

Davies' Analytical Geometry. \ The original compendiums, for those de- 

Davies' Diff. *fc Int. Calculus, f siring to give full time to each branch. 

Davies ' Descrintive Geometry.— With application to Spherical Trigonome- 
try, Spherical Projections, and Warped Surfaces. 

Daries' Shades, Shadows, and Perspective.— A succinct exposition of 
the mathematical principles involved. 

Davies' Science of Mathematics — For teachers, embracing 

I. Grammar of Arithmetic, UT. Logic and Utility of Mathematics, 
II. Outlines of Mathematics, IV. Mathematical Dictionary. 



Keys may be obtained from the Publishers by Teachers only. 



' 1 



- 



> 



- Copyright, 1858, by Charles Davies. 
Copyright Renewed, 1S8G, by Mary Ann Davjes- 
::"L. GBOM, 



PREFACE. 



Those who are conversant with the preparation cf ele- 
mentary text-books, have experienced the difficulty of 
adapting them to the various wants which they aie in- 
tended to supply. 

The institutions of education are of all grades, from the 
college to the district school, and although there is a wide 
difference between the extremes, the level, in passing 
from one grade to the other, is scarcely broken. 

Each of these classes of seminaries requires text-books 
adapted to its own peculiar wants ; and if each held its 
proper place in its own class, the task of supplying suit- 
able works would not. be difficult. 

An indifferent college is generally inferior, in the system 
and scope of its instruction,to the academy or high school; 
while the district school is often found to be superior to 
its neighboring academy. 

The Geometry of Legendre, embracing a complete 
course oT Geometrical science, is all that is desired in 
the colleges and higher seminaries; while the Practical 
Mathematics for Practical Men, recently published, is 
designed to meet the wants of those schools which are 
strictly elementary and practical in their systems of 
instruction. 



PREFACE 



But still a large class of seminaries remained unsup- 
plied with a suitable text-book on Elementary Geometry 
and Trigonometry : viz., those where the pupils are car- 
ried beyond the acquisition of facts and mere practical 
knowledge, but have not time to go through with a full 
course of mathematical studies. 

It is tor such, that the following work is designed. It 
has been the aim of the author to present the striking 
and important truths of Geometry in a form moie simple 
and concise than could be adopted in a complete treatise 
and yet to preserve the exactness of rigorous reasoning. 

In this system of Geometry nothing has been taken foi 
granted, and nothing passed over without being fully de- 
monstrated. 

The Trigonometry, including the applications to the 
measurements of heights and distances, has been writ- 
ten upon the same plan and for the same objects : it 
embraces all the important theorems and all the striking 
examples. 

In order, however, to render the applications of Ge- 
ometry to the mensuration of surfaces and solids complete 
in itself, a few rules have been given which are not de- 
monstrated. This forms an exception to the general plan 
of the work, but being added in the form of an appendix, it 
does not materially break its unity. 

That the work may be useful in advancing the* interests 
of education, is the hope and ardent wish of the author. 
Fishkill Landing, 
May, 1861 



CONTENTS. 



BOOK I. 

Pagb. 

Definitions and Remarks, ..... 9 — ifl 

Axioms, 16 

Properties of Polygons, - 17 — 37 

BOOK II. 

Of the Circle, 38 

Problems relating to the First and Second Books, - • 53 -68 

BOOK III. 

Ratios and Proportions, ... - 69--S1 

BOOK IV. 

Measurement of Areas and Proportions of Figures, - • 82 — 108 

Problems relating to the Fourth Book, .... 109 — 113 

Appendix — Regular Polygons, - - 113 — 115 

BOOK V 

Of Planea and their Angles, ... 110 — 126 

BOOK VI. 

01 Solids, - 126—102 

Appendix, ....... 163—164 



8, CONTENTS. 




TRIGONOMETRY. 






Page. 


Of Logarithms. - - - - 


165—170 


Of Scales, 


176—181 


Definitions, an i Ex| Janation of Tables, - 


181—189 


Theorems, 


189—192 


Examples, 


193—201 


Application to Heights and Distances, - 


202—210 


APPLICATIONS OF GEOMETRY. 




Mensuration of Surfaces, 


211 


General Principles, 




211—213 


Contents of Figures, .... 




218—289 


Mensuration of Solids, - 




239 


General Principles, .... 




289—240 


Solidities of Figures, ... 




240—241 


Mensuration of the Round Bodies, 




248 


To find the Surface. of a Cylinder, 




248—249 


To find the Solidity of a Cylinder, 




249—250 


To find the Surface of a Cone, 




250—251 


To find the Solidity of a Cone, 




251- -252 


To find the Surface of the Frustum of a Cone, 


263 


To find the Solidity of the Frustum of a Cone, 


254 


To find the Surface of a Sphere, .... 


255 


To find the Surface of a Spherical Zone, 


255—256 


To find the Solidity of a Sphere, 


266—257 


To find the Solidity of a Spherical Segment, - 


258 


To find the Solidity of a Spheroid, - 


269—260 


T: find the Surface of a Cylindrical Ring, 


260—261 


To find the Solidity of a Cylindrical Riq 


B» 


261—262 



ELEMENTARY 
GEOMETRY 



BOOK I. 

DEFINITIONS AND REMARKS. 

1. Extension has three dimensions, length, breadth, ami 
thickness. 

Geometry is the science which has for its object : 

1st. The measurement of extension ; and 2diy, To discover, 

by means of such measurement, the properties and relations 

of geometrical figures. 

2. A Point is that which has place, or position, but not 
magnitude. 

3. A Line is length, without breadth or thickness. 

4. A Straight Line is one which lies 

in the same direction between any two of 

its points. 

5. A Curve Line is one which changes 
is direction at every point. 

The word line when used alone, will designate a straight 
line ; and the word curve, a curve line. 

6. A Surface is that which has length and breadth, with- 
out height or thickness. 

7. A Plane Surface is that which lies even throughout its 
whole extent, and with which a straight line, laid in any 
direction, will exactly coincide in its whole length. 

8. A Curved Surface has length and breadth without thick- 
ness, and like a curve line is constantly changing its direction 

9. A Solid or Body is that which has length, breadth, and 
thickness. Length, breadth, and thickness are called dimen- 



10 GEOMETRY 



De f i ni t i o i s. 



sions. Hence, a solid has three dimensions, a surface two 
and a line one. A point has no dimensions, but position only 

10. Geometry treats of lines, surfaces, and solids. 

1 1 . A Demonstration is a course of reasoning which estab- 
lishes a truth. 

13. An Hypothesis is a supposition on which a demonstra- 
tion may be founded. 

13. A Theorem is something to be proved by demonstration, 

14. A Problem is something proposed to be done. 

15. A Proposition is something proposed either to be done 
or demonstrated — and may be either a problem or a theorem. 

16. A Corollary is an obvious consequence, deduced from 
something that has gone before. 

17. A Scholium is a remark on one or more preceding propo* 
6itions. 

18. An Axiom is a self evident proposition. 

OF ANGLES. 

19. An Angle is the portion of a plane included between 
two straight lines which meet at a common point. The 
tvrD straight lines are called the sides of the angle, and 
the common point of intersection, the vertex. 

Thus, the part of the plane included C 

between AB and A C is called an angle : /^ 

AB and AC are its sides, and A its vertex, a ft 

An angle is generally read, by placing the lettei at the vei 
iox in the middle. Thus, we say, the angle CAB. We may 
however, say simply, the angle A. 

20. One line is said to be perpendicular to another when it 
inclines no more to the one side than to the other 



BOOK 



U 



Definitions, 



The two angles formed are then equal to 
each other. Thus, if the line DB is per- 
pendicular to AC, the angle DBA will 
be equal to DBC. 

21. When two lines are perpendicular 
to each other, the angles which they form 
are called right angles. Thus, DBA and 
DBC are called right angles. 

22. An acute angle is less than a right 
angle. Thus, DBC is an acute angle. 

23. An obtuse angle is greater than a 
right angle. Thus, DBC is an obtuse 
angle. 

24. The circumference of a cirole is a 
curve line all the points of which are 
equally distant from a certain poinJb within 
called the centre. 

Thus, if all the points of the curve AEB 
are equally distant from the centre (7, this 
curve will be the circumference of a circle. 

25. Any portion of the circumference, 
as A ED, is called an arc 

26. The diameter of a circle is a 
straight line passing through the centre 
and terminating at the circumference. 
Thus, A CB is a diameter. 

27. One half of the circumference, as 
ACB is called a semicircumference ; and 
one quarter of the circumference, as A C 
is called a quadrant 



B £ 

D 



B ~C 



b e 



8 rf 





12 



GEOMETRY. 



Definitions. 



28. The circumference of a circle is used for the measure- 
ment of angles. For this purpose it is divided into 360 equal 
parts called degrees, each degree into 60 equal parts .called 
minutes, and each minute into 60 equal parts called seconds. 
The degrees, minutes, and seconds are marked thus ° ' " ; and 
9° 18' 16", are read, 9 degrees 18 minutes and 16 seconds. 

29. Let us suppose the circumference 
of a circle to be divided into 360 degrees, 
beginning at the point B. If through 
the point of division marked 40, we draw- 
ee, then, the angle ECB will be equal to 
40 degrees. If CF were drawn through 
the point of division marked 80, the angle BCF would be equal 
to 80 degrees. 




OF LINES. 

30. Two straight lines are said to be 
parallel, when being produced either way, 
as far as we please, they will not meet 
each other. 

31. Two curves are said to be parallel 
or concentric, when they are the same dis- 
tance from each other at every point. 

32. Oblique lines are those which ap- 
proach each other, and meet if sufficiently 
produced. 

33. Lines which are parallel to the horizon, or to the water 
tevel, are called hor'zontal lines. 

34. Lines which are perpendicular to the horizon, or to the 
water level are called vertical lines 




BOOK 1 



13 



D e fin it io ns. 



OF PLANE FIGURES. 

35. A Plane Figure is a portion of a plane terminated on all 
sides by lines, either straight or curved. 

36. If the lines which bound a figure are straight, the space 
wliich they inclose is called a rectilineal figure, oi polygon 
The lines themselves, taken together, are called the perimeter 
of the polygon. Hence, the perimeter of a poly gen is the sum 
of all its sides. 



37. A polygon of three sides is called 
a triangle. 



38. A polygon of four 
quadrilateral. 



sides is called 




39. A polygon of five sides is called a 
pentagon. 




40. A polygon of six sides is called 
hexagon. 



41. A polygon ot seven sides is called a heptagon 

42 A polygon of eight sides is called an octagon. 
2 



14 



GEOMETRY 



De f in it jo ns. 



43. A polygon of nine sides is called a nonagon. 

44. A polygon of ten sides is called a decagon. 

45. A polygon of twelve sides is called a dodecagon. 

46. There are several kinds of triangles. 



First. An equilateral triangle, which has 
its three sides all equal. 




Second. An isosceles triangle, which has 
two of its sides equal. 



Third. A scalene triangle, 
three sides all unequal. 



rhich has its 





Fourth. A right angled triangle, which 
lias one right angle. 

In the right angled triangle ABC, the 
side AC, opposite the right angle, is called 
the hypothenuse. 

47. The base of a triangle is the side on 
which it stands. Thus, AB is the base of 
the triangle ACB. 

The altitude of a triangle is a line drawn 
from the angle opposite the base and per- ,4 





pendicular to the base, 
angle ACB 



Thus, CD is the altitude of the tri 



BOOK I 



15 



D e f ini ti ons. 



48. There are three kinds of quadrilaterals. 



1. The trapezium, which has none of 
Its sides parallel. 




2. The trapezoid, which has only two 
of its sides parallel. 



8. The parallelogram, which has its 
opposite sides parallel. 



7 



4y. There are four kinds of parallelograms 



1. The rhomboid, which has no right / 

angle. / 



2. The rhombus, or lozenge, which i3 
an equilateral rhomboid. 



3. The rectangle, which is an equian- 
gular parallelogram. 



4. The square, which is both equilat- 
eral and equiangular. 



Iti GEOMETRY 



Of Axioms. 




50. A Diagonal of a figure is a line which 
joins the vertices of two angles not adjacent. 



51. The base of a figure is the side on which it is supposed 
to stand ; and the altitude is a line drawn from the opposite 
side or angle, perpendicular to the base. 

AXIOMS. 

1 . Things wliich are equal to the same thing are equal to 
each other. 

2. If equals be added to equals, the wholes will be equal. 

3. If equals be taken from equals, the remainders will be 
equal. 

4. If equals be added to unequals, the wholes will be un« 
equal. 

5. If equals be taken from unequals, the remainders will be 
unequal. 

6. Things which are double of equal things, are equal to 
each other. 

7. Things which are halves of the same thing, are equal to 
each other. 

8. The whole is greater than any of its parts 

9. The whole is equal to the sum of all its parts. 

10. All right angles are equal to each other. 

11. A straight line is the shortest distance between two 
points. 

12. Magnitudes, which being applied to each other, coin- 
cide throughout their whole extent, are equal. 



BOOK 1 . 



17 



Of Angles. 



PROPERTIES OF POLYGONS. 




THEOREM I. 

Every diameter of a circle divides the circumference into tvn> 
equal parts. 

Let ADBE be the circumference of a 
circle, and ACB a diameter: then will 
the part ADB be equal to the part AEB. 

For, suppose the part AEB to be turn- 
ed around AB, until it shall fall on the 
part ADB. The curve AEB will then 
exactly coincide with the curve ADB, or else there would 
be some point in the curve AEB or A DB, unequally distant 
from the centre C, which is contrary to the definition ol a 
circumference (Def. 24). Hence, the two curves will be 
equal (Ax. 12). 

Corollary 1. If two lines, AB, DE, 
be drawn through the centre C perpen- 
dicular to each other, each will divide the 
circumference into two equal parts ; and 
the entire circumference will be divided 
into the equal quadrants DB, DA, AE, 
and EB. 

Cor. 2. Hence, a right angle, as DCB, is measured by one 
quadrant, or 90 degrees; two right angles by a semicireumfer- 
ence, or ISO degrees ; and four right angles by the whole cir- 
cumference, or 360 degrees 




18 



tx E IYI ETRV, 



Of Angle 




THEOREM II. 

Ij one straight line meet another straight line, the sum of thz 
two adjacent angles will be equal to two right angles. 

Let the straight line CD meet the 
straight line AB, at the point C; then 
will the angle DCB plus the angle DC A 
be equal to two right angles. A C B 

About the centre C, with any radius as CB, suppose a 

semicircumference to be described. Then, the angle DCB 

will be measured by the arc BD, and the angle DC A by the 

arc AD. But the sum of the two arcs is equal to a semicir- 

enmference ■ hence, the sum of the two angles is equal to two 

right angles (Th. i, Cor. 2). 

n 
Coi. 1. If one of the angles, as DCB, 

is a right angle, the other angle, DC A 

will also be a right angle. 

Cor. 2. Hence, all the angles which 
can be formed at any point C\ by any 
number of lines, CD, CE, CF, &c, 
drawn on the same side of AB, are equal 
to two right angles : for, they will be 
measured by a semicircumference. 

Cor. 3. If DC meets two lines CB, CA, making DCB 
plus DC A equal to two right angles, ACB will form one 
straight line. 

Cor. 4. Hence, also, all the angles 
which can be formed round any point, as 
C, are equal to four right angles. For, 
the sum of all the arcs which measure 
them, is equal to the entire circumference, 
which is the measure of four right angles (Th. i. Cor. 2). 





BOOK I 19 



Of Triangles 



THEOREM III. 

Ij two straight lines intersect each other, the opposite or ver- 
tical angles which they form, are equal. 

Let the two straight lines A.B and y 

CD intersect each other at the point 
E : then will the opposite angle A EC — 
be equal to DEB, and AED=CEB. 




For, since the line AE meets the 
line CD, the angle AEC+AED= two right angles. Bui 
since the line DE meets the line AB, we have DEB+AED= 
two right angles. Taking away from these equals the com- 
mon angle AED, and there will remain the angle AEC equal 
to the angle DEB (Ax. 3). 

In the same manner we may prove that the angle AED is 
erjual to the angle CEB. 

THEOREM IV.. 

If two triangles have two sides and the included angle of the 
one, equal to two sides and the included angle of the other, each 
to each, the two triangles will be equal. 

Let the triangles ABC and DEF 
have the side A C equal to DF, CB 
to FE, and the angle C equal to the 
angle F: then will the triangle A CB 
be equal to the triangle DEF. 

For, suppose the side A C, of the A & & 

triangle ACB, to be placed on DF, so that the extremity C 
shall fall on the extremity F: then, since the sides are equal, 
A will fall on D. 

But since the angle C is equal to the angle F, the line CB 




20 GEOMETRY 



Of Tria n g I c 8 




will fall on FE ; and since CB is equal 
to FE, the extremity i? will fall on E ; 
and consequently the side AB will fall 
on the side DE (Ax. 11). Hence, the 
two triangles will fill the same space, 
and consequently are equal (Ax. 12.). ^ # ^ 

Scholium. Two triangles are said to be equal, when being 
applied the one to the other they exactly coincide (Ax. 12). 
Hence, equal triangles have their like parts equal, each to 
each, since those parts coincide with each other. The converse 
of the proposition is also true, namely, that two triangles 
which have all the parts of the one equal to the corresponding 
parts of the other, each to each, are equal : for if applied the 
one to the other, the equal parts will coincide. 

THEOREM V. 

If two triangles have two angles and the included side of tnv 
one, equal to two angles and the included side of the other, each to 
each, the two triangles will be equal. 

Let the two triangles ABC and 
DEF have the angle A equal to the 
angle D, the angle B equal to the 
angle E, and the included side AB 
equal to the included side DE • then 
will the triangle ABC bt equal to the 
triangle DEF. 

For, let the side AB be placed on the side DE, the extrem 
ity A on the extremity D ; and since the sides are equal, the 
point B will fall on the point E. 

Then since *he angle A is equal to the angle D, the side 




BOOK 1. ?*! 



Of Triangles 




AC will take the direction DF: and since the angle B is 
equal to the angle E, the side BC will fall or the side EF : 
hence, the point C will be found at the same time on DF and 
EF, and therefore will fall at the intersection F: consequently, 
all the parts of the triangle ABC will coincide with the parts 
of the triangle DEF, and therefore, the two triangles are equal 

THE O HEM VI. 

In an isosceles triangle the angles opposite the equal sides are 
equal to each other. 

Let ABC be an isosceles triangle, hav- 
ing the side AC equal to the side CB : 
then will the angle A be equal to the an- 
gle B. 

a n b 

For, suppose the line CD to be drawn dividing the angle C 
into two equal parts. 

Then, the two triangles A CD and DCB, have two sides and 
l he included angle of the one equal to two sides and the in- 
cluded angle of the other, each to each : that is, the side AC 
equal to BC, the side CD common, and the included angle 
.4 CD equal to the included angle DCB : hence the two than 
gles are equal (Th. iv) ; and hence, the angle A is equal to 
the angle B. 

Cor. 1. Hence, the line which bisects the vertical angle oi 
an isosceles triangle, bisects the base. It is also perpendicu- 
lar to the base, since the angle CD A is equal to the angle 
CDB. 

Cor. 2. Hence, also, every equilateral triangle, must also 
be equiangular: that is, have all its angles equal, each to each 



22 



GEOMETRY 



Of Triangles 




THEOREM VII. 

Conversely. — If a triangle has two of its angles equal, ifu 
sides opposite those angles will also be equal 

In the triangle ABC, let the angle A be 
fiqual to the angle B : tnen will the side 
BC be equal to the side AC. 

For, if the two sides are not equal, one 
of them must be greater than the other. 
Suppose AC to be the greater side. Then 
take a part AD equal to BC 

Now, in the two triangles ADB and ABC, we have thj 
side ADz=BC, by hypothesis ; the side AB common, and the 
angle A equal to the angle B : hence, the two triangles have 
two sides and the included angle of the one equal to two sides 
and the included angle of the other, each to each : hence, the 
two triangles are equal (Th. iv), that is, a part ADB is 
equal to the whole ABC, which is impossible (Ax. 8) : conse- 
quently, the side A C cannot be greater than the side CB, and 
hence, the triangle is isosceles. 

Scholium 1. The method of reasoning pursued in the last 
theorem, is called the " reductio ad absurdum," or a proof thai 
leads to a known absurdity. 

Let us analyze this method of reasoning. We wished to 
prove that the two sides AC, CB were equal. We supposed 
them unequal, anl AC the greater — that was an hypothesis 
(See Def. 12). We then reasoned on the hypothesis and 
proved a part equal to the whole, which we know to be false 
(Ax. 8) Hence, we conclude that the hypothesis is untrue, 
because after a correct chain of reasoning it leads to a result 
which we know to be absurd 



BOOK 



2B 



Of Triangles 



Scholium 2. Generally, — If the demonstration is based on 
known principles, previously proved, or admitted in the ax- 
ioms, the conclusion will always be true. But, if the demon- 
stration is based on an hypothesis, (as in the last theorem, thai 
AC was the greater side), and the conclusion is contrary to 
what has been previously proved, or admitted in the axioms 
then, it follows, that the hypothesis cannot be true. 

The former is called a direct, and the latter an indirect 
demonstration. 




THEOREM VIII. 
If two triangles have the three sides of the one cqval to t/tf. 
three sides of the other, each to each, the three angles will aim be 
equal, each to each. 

Let the two triangles ABC, ABD, 
have the side A B equal to the side AB, 
the side A C equal to AD, and the side 
CB equal to DB : then will the corres- 
ponding angles also be equal, viz : the 
angle A will be equal to the angle A, the 
angle B to the angle B, and the angle C 
to the angle D. 

For, suppose the triangles to be joined 
by their longest equal sides A B, and the 
line CD to be drawn. 

Then, since the side A C is equal to AD, by hypothesis, the 
triangle ADC will be isosceles; and therefore, the angle ACD 
will be equal to the angle ADC (Th. vi). In like maimer, 
in the triangle CBD, the side CB is equal to DB : hence, the 
angle BCD is equal to the angle BDC. 

Now, by the addition o! equals, we have 



\ 



24 



GEOMETRY. 



Of Triangles. 




ACD+BCD=ADC+BDC 
that is, the angle ACB=ADB. 

Now, the two triangles ACB and ADB 
have two sides and the included angle of 
the one equal to two sides and the in- 
cluded angle of the other, each to each: hence, the remaining 
angles will be equal (Th. iv) : consequently, the angle CAB 
is equal to BAD, and the angle CBA to the angle ABD. 

Sch. The angles of the two triangles which are equal to 
each other, are those which lie opposite the equal sides. 




THEOREM IX. 

If one side of a triangle is produced, the outward angle is 

greater than either of the inward opposite angles. 

Let ABC be a triangle, having the side 
AB produced to D : then will the outward 
angle CBD be greater than either of the 
inward opposite angles A or C. 

For, suppose the side CB to be bisected at the point E 
Draw AE, and produce it until EF is equal to AE, and then 
draw BF. 

Now, since the two triangles AEC and BEFhave AE— 
EF and EC—EB, and the included angle AEC equal to the 
included angle BEF (Th. iii), the two triangles will be equal 
in all respects (Th. iv) : hence, the angle EBF will be equal 
to the angle C. But the angle CBD is greater than the angle 
CBF, consequently it is greater than the angle C. 

In like manner, if CB be produced to G, and AB be bi 
sected, it may be proved that the outward angle ABG, or its 
equal CBD (Th. iii), is greater than the angle A 



BOOK l. 2b 



Of Triangles 




THEOREM X. 

The sum of any two rides of a triangle is greater than th* 
third ride. 

let ABC ha a triangle ■ then will the 
siim of two of its sides, as AC, CB, be 
g eater than tlte third side AB. 

For the straight line AB is the short- 
est distance between the two points A and B (Ax. xi): hence 
AC+CB is greater than AB. 

THEOREM XI. 

The greater side of every triangle is opposite the greater ang^e, 
and conversely, the greater angle is opposite the grca'er side 

First. In the triangle CAB, let the an- 

6 A 

gle C be greater than the angle B : then, 
will the side AB be greater than the side 
A C. ( i T 

For, draw CD, making the angle BCD 
equal to the angle B. Then, the triangle CBD will be 
isosceles : hence, the side CD = DB (Th. vii.) 

Hut, by the last theorem AC is less than AD-\- CD ; thai 
is loss than AD + DB, and consequently less than AB. 

Secondly. Let us suppose the side AB to be greater than 
A C; then will the angle C be greater than the angle B. 

For if the angle C were equal to B, the triangle CA /> 
would re isosceles, and the side A C would be equal to A B 
v Th. vii) , which would be contrary to the hypothesis. 

Again if the angle C were less than B, then, bv the first 
part of the theorem, the side AB would be less than AC, 
which is also contrary to the hypothesis Hence, since C 




2G GEO M E T R Y 



Of Parallel Lines. 



cannot be equal to B, nor less than B, it follows that it must 
be greater 

THEOREM XII. 

// a straight line intersect two parallel lines, the alternate anglis 
will be equal. 

If two parallel straight lines, AB CD, 

are intersected by a third line GH, the ]?/ 

angles AEF and EFD are called alternate " /..*"'" ^ 

angles. It is required to prove that these C ~/P /) 

angles are equal. 

If they are unequal one of them must be greater than the 
other. Suppose EFD to be the greater angle. 

Now conceive FB to be drawn, making the angle EFB 
equal to the angle AEF, and meeting AE in B 

Then, in the triangle FEB the outward angle FEA is greater 
than either of the inward angles B or EFB (Th. ix.) ; and 
therefore, EFB can never be equalto AEFso long as FB meets 
EB. 

But since we have supposed EFD to be greater than AEF, 
it follows that EFB could not be equal to AEF, if FB fell be- 
low FD. Therefore, if the angle EFB is equal to the angle 
AEF, FB cannot meet AB, nor fall below FD, and conse- 
quently must coincide with the parallel CD ( Def. 30) : and 
once, the alternate angles AEF and EFD are equal. 



Cor. If a line be perpendicular to one 
of two parallel lines, it will also be per- 
pendicular to the other 



BOOK!. 27 



Of P a i a I 1 e 1 L i i, c 




THEOREM XIII. 

Conversely, — If a line intersect two straight lines, making ifu 
alternate angles equal, those straight lines will be parallel. 

Let the line EF meet the lines AB, • 

CD making the angle AEF equal to the e/ 

angle EFD: then will the lines AB and 
CD be parallel. C 

For, if they are not parallel, suppose 
through the point F the line FG to be drawn parallel to AB. 

Then, because of the parallels AB, FG, the alternate angles, 
AEF and EFG will be equal (Th. xii). But, by Hypothesis, 
the angle AEF is equal to EFD : hence, the angle EFD is 
equal to the angle EFG (Ax. 1) ; that is, a part is equal to the 
whole, which is absurd (Ax. 8) : therefore, no line but CD can 
be parallel to AB. 

Cor. If two lines are perpendicular to 

the same line, they will be parallel to 

each other. ! 



THEOREM XIV. 

If a line cut two parallel lines, the outward angle is equal to 
the inward opposite angle on the same side; aid the two inward 
angles, on the same side, are equal to two right angles. 

Let the line EF cut the two -parallels 
A 5 CD . then will the outward angle s 

EGB be equa A to the inward opposite an- 



gle EHD ; and the two inward angles, 
BGH and GIID, will be equal to two 
right angles. 




28 GEOMETRY. 



Of Parall-.l L 



First. Since the lines AB, CD, are parallel, the angle AG H 
is equal to the alternate angle GHD E 

(Tli. xii) ; but the angle AGH is equal ^ —ft — ti 

to the opposite angle EGB : hcmce, the _ -^- 

angle EGB is equal to the angle EHD p 
(Ax 1). 

Secondly. Since the two adjacent angles EGB anil BGH 
are equal to two right angles (Th. ii) ; and since the angle 
EGB has been proved equal to EHD, it follows that the sum 
of BGH plus GHD, is also equal to two right angles. 

Cor. 1. Conversely, if one straight line meets two other 
straight lines, making the angles on the same side equal to 
each other, those lines will be parallel. 

Cor. 2. if a line intersect two other lines, making the sum 
of the two inward angles equal to two right angles, those two 
lines will be parallel 

Cor. 3. If a line intersect two other lines, making the sum 
of the two inward angles less than two right angles, those 
lines will not be parallel, but will meet if sufficiently produced. 

THEOREM XV. 

All straight lines which are parallel to the same line, are parallel 
to each other. 

Let the lines AB and CD be each par- q 

ail el to EF: then will they be parallel 
to each other. 

For. let the line Gl be drawn perpen- 
dicular to EF : then will it also be per- 
pendicular to the parallels AB CD (Th. 
uii Cor.). 



A" ~~B 

c D 

E I F 



BOOK I . 



29 



Of Triangles 



Then, since the lines AB and CD are perpendicular to the 
line GI, they will be parallel to each other (I'll. xiii. Cor). 

THEOREM XVI. 

If cue side of a triangle be produced, the outward angle will U 
equal to the sum of the inward opposite angles. 

In the triangle ABC, let the side AB 
be produced to D : then will the outward 
angle CBD be equal to the sum of the in- 
ward opposite angles A and C. 

For, conceive the line BE to be drawn 
parallel to the side AC. Then, since BC meets the two pa- 
rallels AC y BE, the alternate angles A CB and CBE will be 
equal (Th. xii). 

And since the line AD cuts the two parallels BE and AC 
the angles EBD and CAB are equal to each other (Th. xiv) 
Therefore, the inward angles C and A, of the triangle ABC 
are equal to the angles CBE and EBD ; and consequently 
the sum of the two angles, A and C, is equal to the outward 
angle CBD (Ax. 1). 




THEOREM XVII. 



In any triangle the sum of the three anghs is equal to two righ 



angles. 



Let ABC be any triangle: then will 
tne sum of the three angles 

A -\-B-\- C = two right angles. 

For, let the side AB be produced to D 
Then, the outward angle 

CBD --A+C (T\i. xvi) 
3* 




30 



GEOMETRY' 



Of Triangles, 




To each of these equals add the angle 
CBA, and we shall have 

CBD^ CBA=A-\-C+B. 
But the sum of the two angles CBD 
and CBA, is equal to two right angles A 
(Th.ii): hence 

A \- Z»4-C=two right angles (Ax. 1). 

Cor. 1. If two angles of one triangle be equal to two angles 
of another triangle, the third angles will also be equal (Ax. 3). 

Cor. 2. If one angle of one triangle be equal to one angle 
of another triangle, the sum of the two remaining angles in 
each triangle, will also be equal (Ax. 3). 

Cor. 3. If one angle of a triangle be a right angle, the sum 
of the other two angles will be equal to a right angle ; and 
each angle singly, will be acute. 

Cor. 4. No triangle can have more than one right angle, nor 
more than one obtuse angle ; otherwise, the sum of the three 
angles would exceed two right angles : hence, at least two 
angles of every triangle must be acute. 

THEOREM XV] II. 

I. A perpendicular is the shortest line that can be drawn from 
a given point to a given line. 

II. If any number of lines be drawn c rom the same point, those 
which arc nearest the perpendicular are less than those which are 
more remote. 

Let A be a given point, and DE a 
straight line. Suppose AB to be drawn 
perpendiculai to DE, and suppose the 
oblique lines AC and AD also to be 




BOOK J . 



31 



Of Triangles 



drawn : Then, AB will be shorter than either of the oblique 
lines, and AC will be less than AD 

First. Since the angle B, in the triangle ACB, is a righ 
angle, the angle C will be acute (Th. xvii. Cor. 3) : and since 
the greater side of every triangle is opposite the greater angle 
(Th. xi), the side AC will be greater than AB. 

Secondly. Since the angle ACB is acute, the adjacent angle 
ACD will be obtuse (Th. ii) : consequently, the angle D is 
acute (Th. xvii. Cor. 3), and therefore less than the an^rle 
AC J). And since the greater side of every triangle is oppo- 
site the greater angle, it follows that AD is greater than AC. 

Cor. A perpendicular is the shortest distance from a point 
to a line. 



THEOREM XIX. 




G C E 



if two right angled triangles have the hypothenuse and a stiL 
of the one equal to the hypothenuse and a side of the other, the. 
remaining parts will also be equal, each to each. 

Let the two right angled triangles A [) 

ABC and DEF, have the hypothe- 
nuse AC equal to DF, and the side 
AB equal to DE : then will the re- 
maining parts be equal, each to each. " 

For, if the side BC is equal to EF, the corresponding an- 
gles of the two triangles will be equal (Th. viii). If the sides 
are unequal, suppose BC to be the greater, and take apart, 
BG, equal to EF, and draw AG. 

Then, in the two triangles ABG and DEF. the angle B is 
equal to the angle E, the side AB to the side DE, and the side 
TIG to the side EF : hence, the two triangles are equal in all 
respects (Th. iv) and consequently, the side AG is equal to 



32 GEOMETRY 



Of Polygons 



DF. But DF is equal to AC, by hypothesis; therefore. 
AG is equal to AC (Ax. 1). But this is impossible (TL 
xviii) ; hence, the sides BC and EF cannot be unequal ; con- 
sequently, the triangles are equal (Th. viii). 

THEOREM XX 

The sum of the four angles of every quadrilateral is equal to four 

right angles. 

Let A CBD be a quadrilateral : then will .^ 

A + B+C + D=z four right angles. /'[\ 

l^et the diagonal DC be drawn dividing 4/ I : fi 

the quadrilateral AB, into two triangles, ^\l/ 

BDC, ADC. C 

Then, because the sum of the three angles of each triangle 
is equal to two right angles (Th. xvii), it follows that the sum 
of the angles of both triangles is equal to four right angles. 
But the sum of the angles of both triangles, make up the angles 
of the quadrilateral. Hence, the sum of the four angles of the 
quadrilateral is equal to four right angles 

Cor. 1. If then three of the angles be right angles, the 
fourth angle will also be a right angle. 

Cor. 2. If the sum of two of the tour angles be equal to two 
right angles, the sum of the remaining two will also be equal 
lo two right angles. 

Cor. 3. Since all the angles of a square or rectangle, are 
equal to each other (Def. 48), and their sum equal to four 
right angles, it follows that each angle is equal to one right 
angle. 

THEOREM XXI. 

The .mm of all the interior angles of any polygon is equal to 
twice as many right angles, wanting four, as the figure has 
side • 



BOOK I . 



33 



O f P olygons 




Let ABCDE be any polygon: then will 
the sum of its inward angles 

A+B+C+D+ E 
be equil to twice as many right angles, 
wanting four, as the figure has sides. 

For, from an} point P, within the poly- A B 

gon, draw the lines PA, PB, PC, PD, PE, to each cf the 
angles, dividing the polygon into as many triangles ^s the 
figure has sides. 

Now, the sum of the three angles of each of these triangles 
is equal to two right angles (Th. xvii) : hence, the sum of the 
angles of all the triangles is equal to twice as many right an- 
gles as the figure has sides. 

But the sum of all the angles about the point P is equal to 
four right angles (Th. ii. Cor. 4) ; and since this sum makes 
no part of the inward angles of the polygon, it must bfc sub- 
tracted from the sum of all the angles of the triangles, before 
found. Hence, the sum of the interior angles of the polygon 
is equal to twice as many right angles, wanting four, as the figure 
has sides. 



Sch. This proposition is not applicable 
to polygons which have re-entrant angles. 

The reasoning is limited to polygons 
with salient angles, which may properly 
be named convex polygons. 




THEOREM XXII. 



[f every side of a polygon be produced out, the sum of all the oui 
ward angles thereby formed, will be equal to four righi angles 



(jEOMETRY. 



Of Polygons 




Let A, B, C, D, and E, be the outward 
angles of a polygon formed by producing 
all the sides. Then will 

A 4-B+ C+D + E=four right angles. 

For, each interior angle, plus its exte- 
rior angle, as A + a, is equal to two right 
angles (Th. ii). But there are as many exterior as interior 
angles, and as many of each as there are sides of the polygon : 
hence, the sum of all the interior and exterior angles will be 
equal to twice as many right angles as the polygon has sides. 

But the sum of all the interior angles together with four right 
angles, is equal to twice as many right angles as the polygon 
has sides (Th. xxi) : that is, equal to the sum of all the in- 
ward and outward angles taken together. 

From each of these equal sums take away the inward angles, 
and there will remain, the outward angles equal to four right 
angles (Ax. 3). 



THEOREM XXIII 

The opposite sides and angles of every parallelogram are equal, 
each to each : and a diagonal divides the parallelogram into two 
equal triangles. 

Let ABCD be any parallelogram, and 
DB a diagonal : then will the opposite 
sides and angles be equal to each other, 
each to each, and the diagonal DB will 
divide the parallelogram into two equal 
triangles. 

For, since the figure is a parallelogram, the sides AB, DC 
are parallel, as also the sides AD, BC. Now, since the 




BOOK 1. 30 



Of Parallelograms. 



parallels are cut by the diagonal DB, the alternate angles will 
be equal (Th. xii) : that is the angle 

ADB^DBC and BDC=ABD. 

Hence the two triangles ADB BDC, having two angles in 
the one equal to two angles in the other, will have their third 
angles equal (Th. xvii. Cor. 1), viz. the angle A equal to the 
angle C, and these are two of the opposite angles of tho 
parallelogram. 

Also, if to the equal angles ADB, DBC, we add the equals 
BDC, ABD, the sums will be equal (Ax. 2) : viz. the wholo 
angle ADC to the whole angle ABC, and these are the other 
two opposite angles of the parallelogram. 

Again, since the two triangles ADB, DBC, have the side 
DB common, and the two adjacent angles in the one equal to 
the two adjacent angles in the other, each to each, the two 
triangles will be equal (Th. v) : hence, the diagonal divides 
the parallelogram into two equal triangles. 

Cor. 1. If one angle of a parallelogram be a right angle, 
each of the angles will also be a right angle, and the parallelo- 
gram will be a rectangle. 

Cor. 2. Hence, also, the sum of either two adjacent angles 
of a parallelogram, will be equal to two right angles. 



THEOREM XXIV. 

If the opposite sides of a quadrilateral, are equal, each to each 
ike equal sides will be parallel, and the figure will be a pa? 
rallelogram. 



36 



GEOMETRY 



O 1 Parallelograms 




Let ABCD be a quadrilateral, having 
its opposite sides respectively equal, viz. 
AB=CD and AD=BC 

then will these sides be parallel, and the 
figure will be a parallelogram. 

For, draw the diagonal BD. Then, the two triangles A BI) 
BDC, have all the sides of the one equal to all the sides of 
the other, each to each : therefore, the two triangles are equal 
(Th. viii) ; hence, the angle ADB, opposite ^the side AB, is 
equal to the angle DBC opposite the side DC ; therefore, the 
sides AD, BC, are parallel (Th. xiii). For a like reason DC 
is parallel to AB, and the figure ABCD is a parallelogram. 




THEOREM XXV. 
If two opposite sides of a quadrilateral are equal and parallel t 
'lw remaining sides will also be equal and parallel, and the figure 
will be a varallelogram. 

Let ABCD be a quadrilateral, having 
the sides A B, CD, equal and parallel : 
then will the figure be a parallelogram. 

For, draw the diagonal DB, dividing 
the quadrilateral into two triangles. Then, 
since AB is parallel to DC, the alternate angles, ABD and 
BDC are equal (Th. xii) : moreover, the side BD is common ; 
hence the two triangles have two sides and the included angle 
of" the one, equal to two sides and the included angle of the 
O her : the triangles are therefore equal, and consequently 
AD is equal to BC, and the angle ADB to the angle DBC 
and consequently, AD is also parallel to BC (Th xii;) 
Therefore, the figure ABCD is a parallelogram. 



BOOK I 



37 



Of Parallelograms 



THEOREM XXVI. 

Th two diagonals of a parallelogram divide each other into equa. 
parts, or mutually bisect each other 

Let ABCD be a parallelogram, ai>d 
A C, BD its two diagonals intersecting at 
E. Then will 

AE = EC and BE = ED. 



D 




A B 

Comparing the two triangles AED and 
BEC, we rind the side AD = BC (Th. xxiii), the angle 
ADE=EBC and EAD — ECB: hence, the two triangles are 
equal (Th. v) : therefore, AE, the side opposite ADE, u 
equal to EC, the side opposite EBC; and ED is equal to EB 

Sch. In the case of a rhombus (Def. 48), 
the sides AB, BC being equal, the trian- 
gles AEB and BEC have all the sides of 
the one equal to the corresponding sides 
of the other, and are therefore equal. A 
Whence it follows that the angles AEB 
and BEC are equal. Therefore, the diagonals o( a rhomb n 
bisect each other at right angles. 




GEOMETRY 



BOOK II, 



OF THE CIRCLE 



DEFINITIONS. 



1. The ciicumference of a circle is a curve line, all the 
points of which are equally distant from a certain point within 
called the centre. 

2. The circle is the space bounded by this curve line. 

3. Every straight line, CA, CD, CE, drawn 
from the centre to the circumference, is 
called a radius or semidiamster. Every 
line which, like AB, passes through the 
centre and terminates in the circumfe- 
rence, is called a diameter. 




4. Any portion of the circumference, 
as EFG, is called an arc. 

5. A straight line, as EG, joining tho^ 
extremities of an arc, is called a chord. 

6 A segment is the surface or portion 
of a circle included between an arc and 
its chord. Thus EFG is a segment. 




BOOK II. 



3^ 



Definitions 



7. A sector is tue part of the circle in- 
cluded between an arc and the two radii 
diawn through its extremities. Thus. 
CAB is a sector 



8. A straight line is said to be in- 
scribed in a circle, when its extremities 
are in the circumference. Thus, the 
line AB is inscribed in a circle. 



9. An inscribed angle is one which 
is formed by two chords that intersect 
each other in the circumference. Thus, 
BAC is an inscribed angle. 



1U. An inscribed triangle is one 
which has its three angular points in 
the circumference. Thus, ABC is an 
inscribed triangle. J) 




11. Any polygon is said to be in- 
scribed in a circle when the vertices of 
all the angles are in the circumference. 
The ciicle is then said to circumscribe 
the polygon. 




40 



GEOMETRY 



Definitions 



12 A secant is a line which meets the 
circumference in two points, and lies 
partly within and partly without the 
Thus A B is a secant. 



circle. 



13. A tangent is a line which has 
but one point in common with the cir- 
cumference. Thus, CMB is a tangent. \ 




14. Two circles are said to touch 
each other internally, when one lies 
within the other, and their circumfe- 
rences have but one point in common. 




15. Two circles are said to touch 
each other externally, when one lies 
without the other, and their circumfe- 
rences have but one point in common 




BOOK II. 



41 



Of the Circle, 



THEOREM I. 

A diameter is greater than any other chord. 
Let AD be any chord. Draw 



the radii CA, CD to its extremities. 
We shall then have A C-\- CD greater 
than AD (Book I. Th. X*). But A 
AC+CD is equal to the diameter 
AB : hence, the diameter AB is 
creator than AD. 




THEOREM [I. 

If from the centre of a circle a line be drawn to the middle of 
a chord, 

I . It will be perpendicu \ar to the chord ; 

II. And it will bisect the arc of the chord. 
Let C be the centre of a circle, and 

AB any chord. Draw CD through 
D, the middle point of the chord, and 
produce it to E : then will CD be 
perpendicular to the chord, and the 
arc AE equal to EB. 

First. Draw the two radii CA, CB. 
Then the two triangles A CD, D CB, 
have the three sides of the one equal to the three sides of the 




*Note. When reference is made from one theorem to unother, in the 
same Book, the number of tne theorem referred to is alone given • but 
when the theorem referre 1 to is found ir a preceding Book, the number of 



the Book is also g ; ven. 
4* 



42 



G E M £ T R V 



Of the Circle. 




other, each to each: viz. AC equal to 
CB, being radii, AD equal to DB, by 
hypothesis, and CD common: hence, 
the corresponding angles are equal 
(Hook I. Th, viii) : that is, the angle 
CD A equal to CDB, and the angle 
ACD equal to the angle DCB. 

But, since the angle CD A is equal 
to the angle CDB, the radius CE is perpendicular to tbe 
chord AB (Bk. I. Def. 20). 

Secondly. Since the angle ACE is equal to BCE, the 
arc A E will be equal to the arc EB, for equal angles must 
have equal measures (Bk. I. Def. 29). 

Hence, the radius drawn through the middle point of a chord, 
is perpendicular to the chord, and bisects the arc of the chord. 

Cor. Hence, a line which bisects a chord at right angles, 
bisects the arc of the chord, and passes through the centre of 
the circle. Also, a line drawn through the centre of the cir- 
cle and perpendicular to the chord, bisects it. 

THEOREM III. 

If more than two equal lines can be drawn from any point witfun 
a circle to the circumference, that point will be the centre. 

Let D be any point within the circle 
ABC. Then, if the three lines DA, 
DB, and DC, drawn from the point D 
to the circumference, are equal, the 
point D will be the centre. 

For, draw the chords AB, BC, bi- 
sect them at the points E and F, and 
ioin DE and DF. 




BOOK II 



43 



Of the Circle 



Then, since the two triangles DAE and DEB have the side 
AE equal to EB, AD equal to DB, and DE common, they 
will be equal in all respects ; and consequently, the angle 
DEA is equal to the angle DEB (Bk. I. Th. viii) ; and 
therefore, DE is perpendicular to AB (Bk. I. De[. 20) But 
if DE bisects AB at right angles, it wiL pass through the 
centre of the circle (Th. ii. Cor). 

In like manner, it may be shown that DF passes through 
the centre of the circle, and since the centre is found in the 
two lines ED, DF, it will be found at their common inter- 
section D. 



THEOREM IV. 

Any chords which are equally distant from the centre of a circle, 
are equal. 

Let AB and ED be two chords equally 
distant from the centre C : then will the 
two chords AB, ED be equal to each 
other 

Draw CF perpendicular to AB, and 
CG perpendicular to ED, and since these 
perpendiculars measure the distances from 
the centre, they will be equal. Also draw 
CB and CE. 

Then, the two right angled triangles CFB and CEG hav 
ing the hypothenuse CB equal to the hypothenuse CE, and 
the side CF equal to CG, will have the third side BF equal U: 
EG (Bk. I Th. xix) But, BF is the half of BA and EG 
the half oi^ DE (Th. ii. Cor); hence BA is equal to DE 
(Ax 6). 




44 



G E O M E T RY. 



Of the Cir 




THEOREM V. 

A line which is perpendicular to a radius at its extremity, is 
tangent to the circle. 

Lot the line ABD be perpendicular 
to the radius CB at the extremity B : 
then will it be tangent to the circle at 
the point B. 

For, from any other point of the 
line, as D, draw DFC to the centre, 
cutting the circumference in F. 

Then, because the angle B, of the 
triangle CDB, is a right angle, the angle at D is acute (Bk. 1. 
Th. xvii. Cor. 3), and consequently less than the angle B 
But the greater side of every triangle is opposite to the greater 
angle (Bk. I. Th. xi) ; therefore, the side CD is greater than 
CB, or its equal CF. Hence, the point D is without the cir- 
cle, and the same may be shown for every other point of the 
line AD. Consequently, the line ABD has but one point in 
common with the circumference of the circle, and therefore 
is tangent to it at the point B (Def. 13) 

Cor. Hence, if a line is tangent to a circle, and a radius be 
drawn through the point of contact, the radius will be perpen 
dicular to the tangent. 



THEOREM VI. 



if the distance betvieen the centres of two circles is equal to 
the sum of their radii, the two circles will touch each other 
externally. 



BOOK II. 



45 



Of the C irele 




Let C and D be the two centres, and 
suppose the distance between them to 
be equal to the sum of the radii, that is, 
to CA-\ AD 

The circumferences of the circles — "" 

will evidently have the points common, and they will have n » 
other. Because, if they had two points common, that, is if they 
cut each other in two points, G and H, the distance CD be- 
tween their centres would be less than the sum of their radii 
CH, HD (Bk. I. Th. x) ; but this would be contrary to the 
supposition. 




THEOREM VII. 

If the distance between the centres of two circles ts equal i 
the difference of their radii, the two circles will touch each otb 
internally. 

Let C and D be the centres of two 
circles at a distance from each other 
equal to AD— AC— CD. Fr 

Now, it is evident, as in the last theo- \ V 
rem, that the circumferences will have the 
point A common ; and they can have no 
other. For, if they had two points common, the difference be- 
tween the radii AD and FC would not be equal to CD, thv 
distance between their centres : therefore, they cannot have 
two points in common when the difference of their radii is 
equal to the distance between their centres : hence, they are 
tangent to each other. 

Sch If two circles touch each other, either externally 01 
internally, their centres and the point of contact will be in the 
same straight line 



46 



GEO M E T K V 



Of the Circle 



THEOREM VIII 



An angle at the circumference of a circle is measured by half the 
arc that subtends it 




Lt-\ BAD he an inscribed angle : then 
will it be measured by half the arc BED, 
whuh subtends it. 

For, through the centre C draw the 
diameter ACE, and draw the radii BC, 
CD. 



Then, in the triangie ABC, the exte- 
rior angle BCE is equal to the sum of 
the interior angles B and A (Bk. I. Th. xvi). But since the 
triangle BAC is isosceles, the angles A and B are equal 
(Bk. I. Th. vi) ; therefore, the exterior angle BCE is equal 
to double the angle BA C. 

But, the angle BCE is measured by the arc BE, which 
subtends it ; and consequently, the angle BAE, which is hali 
of BCE, is measured by half the arc BE. 

It may be shown, in like manner, that the angle EAD is 
measured by half the arc ED: and hence, by the addition of 
equals, it would follow that, the angle BAD is measured by 
half the arc BED, which subtends it. 

Cor. 1. Hence, if an angle at the centre, and an angle it the 
circumference, both stand on the same arc, the angle at the 
ccntie will be double the angle at the circumference. 



Cor. 2. If two angles at the circumference stand on equal 
arcH thev will be equal to each other. 



BOOK II. 



47 



Of the Circle 



• THEOREM II. 

A 11 angles at the circumference, which, stand upon the same arc 
are equal to each other. 

Let the angles BAC, BDC, BFC, have 
their vertices in the circumference, and 
stand on the same arc BEC : then will 
they be equal to each other. 

For, each angle is measured by half ^ 
the arc BEC (Th. viii) ; hence, the an- 
gles are all equal. E 




B 



THEOREM X. 

An angle in a semicircle, is a right angle. 

Let ABBC be a semicircle : then will 
every angle, as B, B, inscribed in it, be 
a right angle. 

For, each angle is measured by half j 
the semic^rcumference ADC, that is, by a 
quadiant, which measures a right angle 
fBk I. Th. i. Cor. 2). D 



B 




THEOREM XI. 

If a quadrilateral be inscribed in a circle, the sum of either tiro 
of its opposite angles is equal to two right angles. 

Let A BCD be any quadrilateral in- 
scribed in a circle ; then will the sum of 
the two opposite angles, A and C, or B 
and D, be equal to two right angles. u 

For, the angle A is measured by half 
the arc DCB, which subtends it (Th. viii) ; 




48 



G E O M E T R Y . 



Of the Ci re! 



and the angle C is measured by half the 

arc DAB, which subtends it. Hence, 

the sum of the two angles, A and C is 

measured by half the entire circumference. I\ 

But half the entire circumference is the n 

measure of two % right angles; therefore, 

th? sum of the opposite angles A and C is equal 

angles. 

In like manner, it may be shown, that the 
wo angles B and D is equal to two right angles 




sum 



THEOREM XII. 

If the side of a quadrilateral, inscribed m a circle, be pro- 
duced out, the exterior angle will be equal to the inward opposite 
angle 

Let the side BA, of the quadrilateral 
A BCD be produced to E, then will the 
outward angle DAE be equal to the in- 
ward opposite angle C. 

For, the angle DAB plus the angle C, 
ts equal to two right angles (Th. xi). But 
DAB plus DA E is also equal to two right angles (Bk. I. Th. ii). 
Taking from each the common angle DAB, and we shall have 
the angle DAE equal to the interior opposite angle C. 




THEOREM XIII. 

Two parallel chords intercept equal arcs. 



BOOK II. 



49 



Of the Circle 



Let the chords AB and CD be parallel: 
then will the arcs AC and BD be equal. 

For, draw the line AD. Then, because 
the lines AB and CD are parallel, the 
alternate angles ADC and DAB will be 
equal (Bk. I. Th. xii). But the angle 
ADC is measured by half the arc AC, 
and the angle DAB by half the arc BD (Th. viii) : 
the two arcs A C and BD are themselves equal. 



henct 



THEOREM XIV. 

The angle formed by a tangent and a chord, is measured by half 
the arc of the chord. 

Let BAE be tangent to the circle at the 
point A, and AC any chord. 

From A, the point of contact, draw the 
diameter AD. 

Then, the angle BAD will be a right 
angle (Th. v. Cor), and therefore will be 
measured by half the semicircle AMD B 
(Bk. I, Th. i. Cor. 2). 

But the angle DA C being at the circumference, is measure 1 
by half the arc DC: hence, by the addition of equals, the two 
angles BAD and DAC, or the entire angle BAC will be meas- 
ured by half the arc AMDC. 

It may be shown, by taking the difference between the two 
angles DAE and DAC, that the angle CAE is measured by 
half the arc AC included between its sides. 
5 




60 



GEOMETRY 



Of the Circl 



THEOREM XV. 



If a tangent and a chord are parallel to each other , they will 
intercept equal arcs. 

Let the tangent ABC be parallel to the 
chord DF : then will the intercepted arcs 
DB, BF, be equal to each other. 

For, draw the chord DB. Then, since 
AC and DF are parallel, the angle ABD 
will be equal to the angle BDF. But 
ABD being formed by a tangent and a 
chord, will be measured by half the arc 
DB ; and BDF being an angle at the circumference will be 
measured by half the arc BF (Th. viii). But since the angles 
?ire equal, the arcs will be equal : hence DB is equal to BF. 




THEOREM XVI 

The angle formed within a circle by the intersection of two 
chords, is measured by half the sum of the intercepted arcs. 

Let the two chords AB and CD inter- 
sect each other at the point E : then will 
the angle AEC, or its equal DEB, be 
measured by half the sum of the inter- 
cepted arcs AC, DB. 

For, draw the chord AF parallel to 
CD. Then because of the parallels, the 
angle DEB will be equal to the angle FAB (Bk I. Th. xiv), 
and the arc FD to the arc AC. But the angle FAB is meas- 
ured by half the arc FDB, that is, by half the sum of the arcs 
FD, DB. Now, since FD is equal to AC, it follows that the 
angle DEB, or its equal AEC, will be measured by half the 
sum of the arcs DB arifl A C 




BOOK II. 



51 



Of the Circle. 



THEOREM XVII. 

The angle formed without a circle by the intersection cf 
two secants is measured by half the difference of the intercepted 
arcs. 

Let the two secants DE and EB inter- 
sect each other at E : then will the angle 
DEB be measured by half the intercepted 
arcs CA and DB. 

Draw the chord AF parallel to ED. D> 
Then, because AF and ED are parallel, 
and EB cuts them, the angles FAB and 
and DEB are equal (Bk. I. Th. xiv). 

But the angle FAB. at the circumference, is measured by 
half the arc FB (Th. viii), which is the difference of the arcs 
DFB and CA : hence, the equal angle E is also measured by 
half the difference of the intercepted arcs DFB and CA 




THEOREM XVIII. 

An angle formed by two tangents is measured by half the 
difference of the intercepted arcs. 

Let CD and DA be two tangents to 
the circle at the points C and A : then 
will the angle CD A be measured by half 
the difference of the intercepted arcs CEA 
and CFA. 

For, draw the chord AF parallel to the 
tangent CD. Then, because the lines 
CD and AF are parallel, the angle BAF 
will be equal to the angle BDC (Bk. I. Th. xiv). But the 
ang-le BAF, formed by a tangent and a chord, is measured by 




52 



G E O ;f E T R V 



Of the Circle 



half the aic AF, that is, by half the 
difference of CFA and CF. 

But since the tangent DC and the 
chord A F are parallel, the arc CF is 
equal to the arc CA : hence the angle 
BAF, or its equal BDC, which is meas-/ 
ured by half the difference of CFA and 
CF, is also measured by half the differ- 
ence of the intercepted arcs CFA and CA. 



Ccr. In like manner it may be proved 
that the angle E, formed by a tangent and 
secant, is measured by half the difference 
of the intercepted arcs AC and DBA. 




THEOREM XIX 

The cJiOrd of an arc of sixty degrees is equal to the radius of 
the circle. 

" et AEB be an arc of sixty degrees 
and AB its chord: then will AB be equal 
to the radius of the circle. 

For, draw the radii CB and CA. 
Then, since the angle ACB is at the 
centre, it will be measured by the arc 
AEB: that is, it will be equal to sixty 
degrees (Bk. I. Def. 29). 

Again, since the sum of the three angles of a triangle is 
equal to one hundred and eighty degrees (Bk. I. Th. xvii), it. 




BOOK II. 



53 



Of the Circle. 



follows that the sum of the two angles A and B will be equal 
to one hundred and twenty degrees. But the triangle CA B 
is isosceles: hence, the angles at the base are equal (Bk. I. 
1 h. vi) : hence, each angle is equal to sixty degrees, and 
consequently, the side AB is equal to AC or CB (Bk. I. Th vi) 



PROBLEMS 



RELATING TO THE FIRST AND SECOND BOOKS. 



The Problems of Geometry explain the methods of con 
structing or describing the geometrical figures. 

For these constructions, a straight ruler and the common 
compasses or dividers, are all the instruments that are ab- 
solutely necessary. 

DIVIDERS OR COMPASSES. 




The dividers consist of the two legs ha, be, which turn 
easily about a common joint at b. The legs of the dividers 



64 



GEOMETRY 



Problems 



are extended or brought together fry placing the forefinger on 
the joint at b, and pressing the thumb and fingers against the 
legs 



PROBLEM 1. 
On my line, as CD, to lay off a distance equal to A R. 

Take up the dividers with the 
thumb and second finger, and place 
the forefinger on the joint at b. A B 

Then, set one foot of the dividers ~ 
at A, and extend the legs with the ' 
thumb and fingers, until the other 
foot reaches B. 

Then, raise the dividers, place one foot at C, and mark 
with the other the distance CE : and this distance will evi- 
dently be equal to AB. 



E D 



PROBLEM II. 

To describe from a given centre the circumference of a circle 
having a given radius. 

Let C be the given centre, and 
CB the given radius. 

Place one foot of the dividers at 
C and extend the other hg until it 
reaches to B. Then, turn the di- 
viders around the leg at C, and the 
othei leg will describe the required 
circumference 




BOOK II. 



53 



Problems. 



OF THE RULER. 




A ruler of a convenient size, is about twenty inches in 
length, two inches wide, and one fifth of an inch in thickness. 
It should be made of a hard material, and perfectly straighl 
and smooth. 



PROBLEM III. 

To draw a straight line through two given points A and B. 

Place one edge of the ruler on 
A and slide the ruler around until 
he same edge falls on B. Then, 
with a pen, or pencil, draw the 
ine AB. 



B 



TROBLEM IV. 

To bisect a given line : that is, to divide it into two equal parts. 

Let A B be the given line to be 
divided. With iasa centre, and 
radius greater than half of AB, 
describe an arc IFE. Then, with 
Basa centre, and an equal radius 
BI describe the arc IHE. Join 
the points / and E by the line IE . 
the point D, where it intersects 
AB, will be the middle point of the 
line AB. 




£N. 



56 



G E O M E T R 



Problems. 



For, draw the radii AI, AE 
BI, and BE. Then, since these 
radii are equal, the triangles A IE 
pnd B1E have all the sides of the 
one equal to the corresponding sides 
of the other ; hence, iheir corres 
ponding angles are equal (Bk I. 



Th. viii) ; that is, the angle A IE is equal to ihe angle Bl F. 
Therefore, the two triangles AID and BID, have the sidt 
AI—IB, the angle AID = BID, and ID common: lionet 
thev are equal (Bk. I. Th. iv), and AD is equal to DB. 





PROBLEM V. 
To bisect a given angle or a given aic. 

Let A CB be the given angle, 
and AEB the given arc. 

From the points A and B, as 
centres, describe with the same 
radius two arcs cutting each other 
in D. Through D and the centre 
C, draw CED, and it will divide 
the angle ACB into two equal parts, and also bisect the arc 
AEB bx£. 

For, draw the radii AD and BD. Then, in the two triangles 
A CD, CBD, we have 

AC=CB, AD = BD 

and CD common : hence, the two triangles have their corres- 
ponding angles equal (Bk I. Th. viii), and consequently, A CD 
is equal to BCD. But since A CD is equal to BCD, it fol 
lows that the arc AE, which measures the former, is equal tc 
the arc BE. which measures the latter 



BOOK 11. 



57 



Problems. 



PROBLEM VI. 



At a given point in a straight line tc erect a perpendicular to ttu 

line. 



Let A be the given point, and BC 
the given line. 

From, A lay off any two distances, 
A B and A C, equal to each other 
Then, from the points B and C, as 
centres, with a radius greater than 



JTC--'" 







AB, describe two arcs intersecting each other at. D ; diaw 
DA, and it will be the perpendicular required. 

For, draw the equal radii BD, DC. Then, the two trian- 
gles, BDA, and CD A, will have 

AB—AC BD — DC 

and AD common : hence, the angle DAB is equal to the angle 
DAC (Bk. I. Th. viii), and consequently, DA is perpendicu- 
lar to BC. (Bk. I Def. 21). 



SECOND METHOD. 

WAen the point A is near the extremity of the line. 

Assume any centre, as P, out of 
the given line. Then with P as a 
centre, and radius from P to A, de- 
scribe the circumference of a circle 
Through C, where the circumference 
cuts BA, draw CPD. Then, through 
D, where CP produced meets the 
circumference, draw DA : then will 

DA be perpendicular tu BA, since CAD is an angle in a 
jsemicirclc (Bk. II. Th. x). 




58 



GEOMETRY 



Problems 




'J5T • 



PROBLEM VII. 

From a given point without a straight line tc let fall a perpen 
dicular on the line. 

Let A be the given point, and BD 
the given line 

From the point A as a centre, with 
a radius greater than the shortest 
distance to BD, describe an arc cut- 
ting BD in the points B and D. 
Then, with B and D as centres, and 
the same radius, describe two arcs intersecting each other at 
E. Draw AFE, and it will be the perpendicular required. 

For, draw the equal radii AB, AD, BE and DE Then, 
the two triangles EAB and EAD will have the sides of the 
one equal to the sides of the other, each to each ; hence, their 
corresponding angles will be equal (Bk. I. Th. viii), viz. the 
angle BAE to the angle DAE. Hence, the two triangles 
BAF and DA F will have two sides and the included angle of 
the one, equal to two sides and the included angle of the other, 
and therefore, the angle AFB will be equal to the angle 
AFD (Bk. 1. Th. iv) : hence, AFE will be perpendiculat 
to BD. 

SECOND METHOD 

When the given point A is nearly 
opposite the extremity of the line. 

Draw A C, to any point C of the 
line BD. Bisect AC at P. Then, 
with P as a centre and PC as a ra- 
dius, describe the semiciicle CD A ; 
draw A D, and it will be perpendicular 
to CD since CD A is an angle in a semicircle (Bk. II. Th. x). 




BOOK II. 59 



Problems. 



PROBLEM VIII. 

At a given point in a given line, to make an angle equal to a 
given angle 

Let A be the given point, AE , n 

the given line, and IKL the given y // \ y^\ 

angle. / V / \ 

From the vertex K, as a centre, *- ' J A -& 

with any radius, describe the arc IL, terminating in the two 
sides of the angle : and draw the chord IL. 

From the point A, as a centre, with a distance AE, equaJ 
to KI, describe the arc DE ; then with E, as a centre, and a 
radius equal to the chord IL, describe an arc cutting DE at 
D; draw AD, and the angle EAD will be equal to the 
angle K. 

For, draw the chord DE. Then the two triangles IKL 
and EAD, having the three sides of the one equal to the three 
sides of the other, each to each, the angle EAD will be equal 
to the angle K (Bk. I. Th. viii). 

PROBLEM IX. 

Through a given point to draw a line that shall be parallel to a 
given line. 

Let A be the given point and p p 

BC the given line. 

With A as a centre, and any ra- 

A^- l D 

dius greater than the shortest dis- 
tance from A to BC, describe the indefinite arc DE. From 
the point E, as a centre, with the same radius, describe the 
arc AF : then, make ED equa to A F and draw A D, and it 
wil] be the required parallel. 



bO 



GEOMETRY. 



Problem 



B~ 



For, since the arcs AF and ED 
are equal, the angles EAD and 
AEF, which they measure, are 
equal : hence, the line AD is 
parallel to BC (Bk 1. Th xiii). 



i^ 



E 




PROBLEM X. 

Two angles of a triangle being given or known, to find the. thirl 

Draw the indefinite line 
DEF. 

At any point, as E, make 
the angle DEC equal to one E 

of the given angles, and then CEFI equal to a second, by 
Prob. VIII ; then will the angle HEF be equal to the third 
angle of the triangle. 

For, the sum of the three angles of a triangle is equal to 
two right angles (Bk. I. Th. xvii) ; and the sum of the three 
angles on the same side of the line DE is equal to two right 
angles (Bk. I. Th. ii. Cor. 2) ; hence, if DEC and CEH are 
equal to two of the angles, the angle HEF will be equal to the 
remaining angle of the triangle 

PROBLEM XI. 

Three sides of a triangle being given, to describe the triangle 

Let A, B, and (7, be the given 
sides. 

Draw DE, and make it equal to 

the side A. From the point D, as 

a centre, with a radius equal to the -£" 

B\- 
8P:cond side B. describe an arc O- 




BO K I 1. 61 



\ Problems. 

' from £asa centre, with the third side C, describe another arc 
intersecting the former in F: draw DF and FE: then will 
DEF be the required triangle. 

• For, the three sides are respectively equal to the three lines 
L B, and C. 



PROBLEM XII. 

The adjacent sides of a parallelogram, until Ctie angle uihtrh 'h/n f 
contain, being given, to describe the jmrallelogiain 

Let A and B be the given sides ,, 

and .C the given angle. / 



Draw the line DE and make it ^L 




equal to A. At the point D make A 1 ' 

the angle EDF equal to the angle 

C. Make the side DF equal to B. Then describe two ares, 

one from F as a centre, with a radius FG equal to DE, tht 

other from E, as a centre, with a radius EG equal to DF. 

Through the point 67, the point of intersection, draw the lines 

EG and FG, and D EGF will be the required parallelogram. 

For, in the quadrilateral DFGE, the opposite sides DE 
and FG are each equal to A : the opposite sides DF and 
EG are each equal to B, and the angle EDF is equa. 
lo C. But, since the opposite sides are equal, the}'- arc 
also parallel (Bk. I. Th. xxiv), and therefore the figure is a 
arallelogram 



PROBLEM XIII. 

To describe a square on a given line. 
6 



02 



GEOMETRY 



Problems, 




Let AB be the given line. 

At the point B draw B C perpendicu- 
lar to AB, by Problem VI, and then 
make it equal to AB. 

Then, with A as a centre, and ra- 
dius equal to AB, describe an arc ; and 
with C as a centre, and the same 
radius AB, describe another arc; and through D, their point 
of intersection, draw AD and CD : then will ABCD be the 
required square. 

For, since the opposite sides are equal, the figure will be a 
parallelogram (Bk. I. Th. xxiv) : and since one of the angleo 
is a right angle, the others will also be right angles (Bk- I. 
Th. xxiii. Cor. 1 ) ; and since the sides are all equal, the figure 
will be a square. 



PROBLEM XIV. 

To construct a rhombus, having given the length of one of the 
equal sides, and one of the angles. 

Let AB be equal to the given side, 
and E the given angle. 

At B lay off an angle, ABC, equal 
to E, by Prob. VIII. and make BC 
equal to AB. Then, with A and C 
as centres, and a radius equal to AB, ^ & 

describe two arcs. Through D, their point of intersection, 
draw the lines AD, CD: then will ABCD be the required 
rhombus, 

For, since the opposite sides are equal, they will be parallel 
(Bk. 1. Th. xxiv). But they are each equal to AB. and the 




BOOK II 



fi3 



Problems. 



angle B is equal to the angle E : hence, ABCD is the re- 
quired rhombus. 



PROBLEM xv. 
To find the centre of a circle 

Draw any chord, as AB, and bisect it 
by Problem IV. Then, through F, the 
middle point, draw DCE, perpendicular 
to AB, by Problem VI. Then DCE 
will be a diameter of the circle (Bk. II. 
Th. ii. Cor.). Then bisect DE at C, 
and C will be the centre of the circle. 




PROBLEM XVI. 

To describe the circumference of a circle through three given 
points not in the same straight line. 

Let A y B, C, be the given points. 

Join these points by the straight 
lines AC AB, BC. 

Then, bisect any two of these 
straight lines, as AB, BC, by the 
perpendiculars OD, OP (Prob. iv) ; 
and the point O, where these per- 
pendiculars intersect each other, 
will be the centre of the circle. 

Then with O as a centre, and a radius equal to OA, de« 
scribe the circumference of a circle, and it will pass through 
the points A, B, and C. 

For, the two right angled triangles OAP and OBP have the 
side AP equal to the side BP, OP common, and the included 




64 



GEOMETRY 



P ro b 1 e m* . 



angles OP A and OPB equal, being 
right angles; hence, the side OB is 
equal to OA (Bk. I. Th. iv). 

In like manner it may be shown 
that OC is equal to OB. IJence, a 
circumference described with the 
radius OA, will pass 
points B and C. 



through the 




Sch. This problem enables us to describe the circumference 
of a circle about a given triangle. For, we may consider the 
vertices of the three angles as the three points through which 
the circumference is to pass. 



PROBLEM XVII. 



7krough a given point in the circumference of a circle, to drau 
a tangent line to the circle. 



Let A be the given point 

Through A, draw the radius AC to the 
centre, and then draw DAE perpendicu- 
lar to AC, by Problem VI. Then will 
DAE be tangent to the circle at the point 
A (Bk. II. fh. \) 




PROBLEM XVIII. 



Thrwgh a given point without the circumference, to draw o 
tangent line to the 'circle. 



BOOK 11. 



65 



Problems 



Let C be the centre of the circle, and 
A the given point without the circle. 

Join A and the centre C, and on A C 
as a d'.ameter, describe a circumference. 
'I'h rough ihe points B and D where 
the two circumferences intersect each 
other, draw the lines AB and AD: 
these lines will be tangent to the circle 
»vhose centre is C. 

For, since the angles ABC and 
ADC are each inscribed in a semicircle, they will be right 
angles (Bk. II. Th. x). Again, since the tines AB, AD. 
are each perpendicular to a radius at its extremity, they will 
be tangent to the circle (Bk. II. Th. v). 




PROBLEM XIX 

To inscribe a circle in n given triangle. 

Let ABC be the given tri- 
angle. 

Bisect the angles A and B 
by the lines AO and BO, meet- 
ing at the point 0. From O, 
let fall the perpendiculars OD, 
OE, OF, on the three sides of 
the triangle — these perpendiculars will be equal to each other. 

For, in the two right angled triangles DAO and FAO, we 

ha/e the right angle D equal the right angle F, the angle FAO 

equal to DAO, and consequently, the third angles AOD and 

AOF are equal (Bk. I. Th xvii. Cor I) But the two 

triangles have a common side AO, hence, they are equal 

(Bk. I. Th v), and consequently, OD is equal to OF 
6* 




66 



GEOMETRY 



Problems 



In a similar manner, it may 
be proved that OE and OD arc 
equal . hence, the three per- 
pendiculars, OD, OF, and OE, 
are all equal. 

Now, if with O as a centre,^ 
and OF as a radius, we describe 

the circumference of a circle, it will pass through the points 
D and E. and since the sides of the triangle are perpendiculai 
to the radii OF, OD, OE, they will be tangent to the circum- 
ference (Bk. II. Th. v). Hence, the circle will be inscribed 
in the triangle. 





PROBLEM XX. 

To inscribe an equilateral triangle in a circle. 

Through the centre C draw any diam- 
eter, as ACB. From 5asa centre, with 
a radius equal to BC, describe the arc 
DCE. Then, draw AD, AE, and DE, 
and DAE will be the required triangle. 

For, since the chords BD, BE, are 
^ach equal to the radius CB, the arcs BD, BE, are each equal 
to sixty degrees (Bk. II. Th. xix), and the arc DBE to one 
hundred and twenty degrees ; hence, the angle DAE is equal 
to sixty degrees (Bk. II. Th. viii). 

Again, since the arc BD is equal to sixty degrees, and the 
arc BDA equal to one hundred and eighty degrees, it follows 
that DA will be equal to one hundred and twenty degrees : 
hence, the angle DEA is equal to sixty degrees, and conse- 
quently, the third an^le ADE, is equal to sixty degrees 



BOOK II 



67 



Problems. 



Therefore, the triangle ADE is equilateral (Bk. I. Th. vi 
Cor. 2). 




PROBLEM XXI. 
To inscribe a regular hexagon in a circle. 

Draw any radius, as AC. Then ap- 
ply the radius AC around the circum- 
ference, and it will give the chords AD, 
DE, EF, FG, GH, and HA, which will 
be the sides of the regular hexagon. For, 
the side of a hexagon is equal to the radius (Bk. II. Th. xix), 

PROBLEM XXII. 

To inscribe a square in a given circle. 

Let ABCD be the given circle. 
Draw the two diameters A C, BD, at 
right angles to each other, and through 
the points A, B, C and D draw the 
lines AB, BC, CD, and DA: then 
will ABCD be the required square. 

For, the four right angled triangles, 
AOB, BOC, COD, and DOA are 
equal, since the sides AO, OB, OC, and OD are equal, beinjj 
radii of the circle ; and the angles at O are equal in each 
being right angles: hence, the sides AB, BC, CD, and DA 
are equal (Bk. I. Th. iv). 

But each of the angles ABC, BCD, CD A, DA B, is a right 
angle, being an angle in a semicircle (Bk. II. Th x) : hence, 
the figure ABCD is a square (Bk. I. Def 48) 




68 



GEOMETRY 



Problems 



Sch. If we bisect the arcs AB, 
BC, CD, DA. and join the points, 
we shall have a regular octagon in- 
scribed in the circle. If we again 
bisect the arcs, and join the points of 
bisection, we shall have a regular 
polygon -of sixteen sides. 



^ 




PROBLEM XXIII. 
To describe a square about a given liiclc. 

Praw the diameters AB, DE, at 
right angles to each other. Through 
the extremities A and B draw FA G 
and HBI parallel to DE, and through 
E and D, draw FEH and GDI par- 
allel to AB: then will FGIH be the 
required square. 

For,. since ACDG is a parallelogram, the opposite sides art-: 
equal (Bk. I. Th. xxiii): and since the angle at C is a right angle 
all the other angles are right angles (Bk. I. Th. xxiii. Cor. 1): 
and as the same may be proved of each of the figures CI, CH 
and CF, it follows lhat all the angles, F, G, I, and //, are 
right angles, and that the sides GI, IH, HF, and FG, are 
equal, each being equal to the diameter of the circle. Henc? 
the figure GIB F is a square (Bk I. Def. 48). 




GEOMETRY. 



BOOK III. 

OF RATIOS \ N D PROPORTIONS. 
DEFINITIONS. 

1. Ratio is the quotient arising from dividing one quantity 
by another quantity of the same kind. Thus, if the numbers 
3 and 6 have the same unit, the ratio of 3 to 6 will he. 
expressed by 

3 

And in general, if A and B represent quantities of the same 
kind, the ratio of A to B will be expressed by 

B 

A 
2 If there be four numbers, 2, 4, 8, 16, having such values 
♦.hat the second divided by the first is t-qual to the fourth di- 
vided by the third, the numbers are said to be in proportion. 
And in general, if there be four quantities A, B, C, and D 
having such values that 

B D 
A~C' 

then, A is said to have the same ratio to B, that C has to D , 
or. the ratio of A to B is equal to the ratio of C to D When 



70 GEOMETRY 



Of Ratios and Proportions. 



four quantities have this relation to each other, they are said to 
be in proportion. Hence, the proportion of four quantities 
results from an equality of their ratios taken two and two . 

To express that the ratio of A to B is equal to the ratio 
of C to D, we write the quantities thus : 
A : B :: C : D ; 
and read, A is to B, as G to D. 

The quantities which are compared together are called tne 
terms of the proportion. The first and last terms are called 
the extremes, and the second and third terms, the means. 
Thus, A and D are the extremes, and B and G the means. 

3. Of four proportional quantities, the first and third are 
called the antecedents, and the second and fourth the conse- 
quents ; and the last is said to be a fourth proportional to the 
other three taken in order. Thus, in the last proportion, A 
and C are the antecedents, and B and D the consequents. 

4. Three quantities are in proportion when the first has the 
same ratio to the second, that the second has to the third ; 
and then the middle term is said to be a mean proportional 
between the two other. For example, 

3 : 6 :: 6 : 12 ; 
and 6 is a mean proportional between 3 and 12. 

5. Quantities are said to be in proportion by inversion, or 
inversely, when the consequents are made the antecedents and 
the antecedents the consequents. 

Thus, if we have the proportion 

3 : 6 :: 8 : 16. 

the inverse proportion would be 

6 : 3 :: 16 : 8. 



B*OJK III. 7i 



Of Ratios and Proportion' 



6. Quantities are said to be in proportion by alternation, oi 
alternately, when antecedent is compared with antecedent and 
consequent with consequent. 

Thus, if we have the proportion 

3 : 6 : : 3 : 16, 
the alternate proportion vrould be 

3 : 8 : : 6 : 16. 

7. Quantities are said to be in proportion by composition, 
when the sum of the antecedent and consequent is compared 
cither with antecedent or consequent. 

Thus, if we have the proportion 

2 : 4 : : 8 : 16, 
the proportion by composition would be 

2 + 4 : 4 :: 8+16 : 16; 
that is, 6 : 4 : : 24 : 16. 

8. Quantities are said to be in proportion by division, when 
the difference of the antecedent and consequent is compared 
either with the antecedent or consequent. 

Thus, if we have the proportion 

3 : 9 : : 12 : 36, 
the proportion by division will be 

9-3 : 9 :: 36-12 : 36; 

that is, 6 : 9 : : 24 : 36. 

9. Equimultiples of two or more quantities are the products 
wrhich arise from multiplying the quantities by the same 
number. 

Thus, if we have any two numbers, as 6 ami 5 and multiply 



72 GEOMETRY. 



Of Ratios and Proportions 



them both by any number, as 9, the equimultiples will be 54 
and 45 ; for 

6x9 = 54 and 5x9 = 45. 

A J 30, mxA and mxB are equimultiples of A and 5, the 
common multiplier being m. 

10. Two variable quantities, A and B, are said to be re- 
ciprocally proportional, or inversely proportional, when one 
increases in the same ratio as the other diminishes. When 
this relation exists, either of them is equal to a constant 
quantity divided by the other. 

Thus, if we had any two numbers, as 2 and 4, so related 
to each other that if we divided one by any number we must 
multiply the other by the same number, one would increase 
in the same ratio as the other would dimmish, and their 
product would not be changed. 

THEOREM I. 

If four quantities are in proportion, the product of the two ea 
tremes will be equal to the product of the two means 

If we have the proportion 

A : B : : C : D 

we have, by Def. 2, 

B_D 
A~ C 
and by clearing the equation of fractions, we have 
BC = AD 
Sch The general principle is verified in the proportion 
between the numbers 

2 : 10 : : 12 : 60 

which gives 

2 * 60=10 y 12 = 120 



BOOK III. 73 



Of Ratios and Proportions. 



THEOREM II. 

If four quantities are so related to each other, that the product 
of two of them is equal to the product of the other two ; then 
two of them may he made the means, and the other two the 
extremes of a proportion. 

Let A, B, C, and D y have such values that 

BxC=AxD 
Divide both sides of the equation by A and we have 

**C=D 

A 

Then divide both sides of the last equation by C, and we 
have 

B_D 
A~C 
hence, by Def. 2, we have 

A : B : : C : D. 

Sch. The general truth may be verified by the numbers 
2x18 = 9x4 
which give 

2 : 4 : : 9 . 18 

THEOREM III. 

fthiee quantities are in proportion, the product of the two 
extremes will be equal to the square cf the middle term. 

Let us suppose that we have 

A : B . : B : C 
Then, by Def. 2, we have 

B_C 

A~ B 

and by clearing the equation of its fractions, we have 
~1 



74 GEOMETRY. 



Of Ratios and Proportion 



Sch. The proposition may be verified by the numbers 
3 : 6 : : 6 : 12 
flitch givt 

3x12-6x6 = 36 

THEOREM IV. 

If four quantities are in proportion, they will be in proportion 
when taken alternately. 

Let A : B : : C : D 

Then, by Def. 2, we have 

B_D 
A~~C 

Q 

Multiplying both members of this equation by — , we have 

B 

CD 

A~B 
and consequently, 

A : C : : B : D. 

Sch. The theorem may be verified by the proportion 
. 10 : 15 : : 20 : 30 
for, we have, by alternation, 

10 : 20 : : 15 : 30. 

THEOREM V. 

If thei e be two sets of proportions, having an antecedent and 
a consequent in the one, equal to an antecedent and a consequent 
in the other; then, the remaining terms will be proportional 

If we have 
A : B : . C . D, and A : B : R : F ; 
then we shall have 



BOOK III . y 6 



Of Ratios and Propor ions. 

B D B F 

2 = C and A=£ 

Hence, by Ax. 1, we have 

D_F 

C^E 

and consequently, 

C : D : : E : F 

Sch. The proposition may be verified by the following 
proportions, 

2 : 6 : : 8 : 24 and 2 : 6 : : 10 : 30 
which give 

8 : 24 : : 10 : 30. 

THEOREM VI. 

If four quantities are in proportion, they will bo in proportion 
when taken inversely. 

If we have the proportion 

A : B : : C : D 
we have, by Th. I, 

AxD=BxC, 
or BxC=AxD. 

Hence, we have, by Th. II, 

B : A : : D : C. 
Sch. The proposition may be verified by the proportion 
7 : 14 : : 8 : 16; 
which, when taken inversely, gives 

14 : 7 : : 16 : 8. 

THEOREM VII. 

4./ four quantities are in proportion, they will be in proportion by 
composition. 



76 GEOMETRY". 



Of R,atios and Propoi ions 



Let us suppose that we hue 

A : B : : C : D 
we shaL then have 

AxD=.-BxC. 
To each of these equals, add BxD, and we have 
(A+B)xD=z(C+D)xB; 
and by separating the factors by Th. II, we have 
A + B : B : : C+D : D. 

Sch. The proposition may be verified by the following 
proportion, 

9 : 27 : : 16 : 48. 
We shall have, by composition, 

9+27 : 27 : : 16 + 48 : 48, 
that is, 36 : 27 : : 64 : 48 

in which the ratio is three fourths. 

THEOREM VIII. 

If four quantities are in p? oportion, they urill be in proportion by 
division. 

Let us suppose that we have 

A : B : : C : D , 
we shall then have 

AxD=BxC. 
From each of these equals let us subtract BxD, and we 
have 

(A-B)xD={C-D)xB; 
and by separating the factors by Th. II, we have, 
A-B : B : : C-D : D. 

Sch The proposition may be verified by the proportion, 

24 • 8 : : 48 : 16 



BOOK III. 77 

Of Ratios and Pro port ions 

We have, by division, 

24-8 : 8 : : 48-16 : 16; 
that is, 16 : 8 : : 32 : 16; 

in which the ratio is one -half. 

THEOREM IX. 

Equal multiples of two quantities have the same ratio as thf 
quantities themselves. 

It wo havo the proportion 

A : B ■ : C : D 
we shall have 

B_D 

A~C 

Now, let M be any number, and by it multiply the nu« 
merator and denominator of the first member of the equation 
which will not change its value : we shall then hn\ e 

MxB D 



MxA C 
and hence we have 

MxA : MxB :: C : D, 
that is, the equal multiples Mx A and MxB, have the same 
ratio as A to B. 

Sch The proposition may be verified by the proportion, 
5 : 10 : : 12 ■ 24; 

for, by multiplying the first antecedent and consequent by any 
number, as 6, we have 

30 : 60 : : 12 : 24, 
v Arhich the ratio is still 2. 

7* 



78 GEOMETRY 



Of Ratios and Proportion 



THEOREM X. 

If four quantities arc proportional, and one antecedent and lis 
consequent be augmented by quantities which have the same ratio 
as the antecedent and consequent, the four quantities will still bs 
in proportion 

Let us take the proportions 
A : B : : C : D, and A : B : : E : F, 
which give 

AxD=zBxC and AxF=BxE; 

adding these equals we have 

Ax{D + F) = Bx{C + E); 
and by Th. II, we have 

A : B : : C+£ : D+F 
in which the antecedent C and its consequent D, are augment- 
ed by the quantities E and F, which have the same ratio. 

Sch. The proposition may be verified by the proportion, 
9 : 18 : : 20 : 40, 
in which the ratio is 2. 

If we augment the antecedent and its consequent bv 1 5 and 
30, which have the same ratio, we have 

9 : 18 : : 20+15 : 404 30 



that is, 9 : 18 : : 35 • 70, 

in which the ratio is still 2. 



THEOREM XI. 



If four quantities are proportional, and one antecedent and its 
consequent be diminished by quantities which have the same ratio 
as the antecedent and consequent, the four quantities will still be 
■in pi'oportion 



BOOK III. 79 
Of Ratios and Proportions. 

Let us take the proportions 
A : B : : G : D, and A : B : : E : F. 
which give 

AxD=BxC and AxF=BxE. 
By subtracting these equalities, we have 
Ax{D-F)=Bx(C-E); 
and by Th. II, we obtain 

A : B : : C-E : D-F, 
m which the antecedent and consequent, C and D, are dimin- 
ished by E and F, which have the same ratio 

Sch. The proposition may be verified by the proportion, 
9 : 18 : : 20 : 40, 
for, by diminishing the antecedent and consequent by 15 and 
30, we have 

9 : 18 :: 20 — 15 : 40 — 30; 

that is 9 : 18 : : 5 : 10 

in which the ratio is still 2. 

THEOREM XII. 

If we have several sets of proportions, having the same ratio, 
any antecedent will be to its consequent, as the sum of the anle< 
cedents to the sum of the consequents. 

If we have the several proportions, 

A : B : : C : D which gives A x D= Bx C 
A : B : : E : F which gives AxF-BxE 
A : B :: G : H which gives AxH=BxG 

We shall then have, by addition, 

Ax{D+F-[-H) = Bx{C-rE+G); 
and consequently, by Th II. 

A : B : : C+E + G : D4-F+H. 



80 GEOMETRY. 



Of Ratios and Proportion 



Sch. The proposition may be verified by the following 
proportions : viz. 
2 • 4 : : G : 12 and 1 : 2 : : 3 : 6 

Then, 2 : 4 : : 6 + 3 : 12 + 6; 

that is, 2 : 4 : • 9 : 18, 

in which the ratio is still 2. 

THEOREM XIII. 

If four quantities are in proportion, their squares or cubes will 
also be proportional. 
If we have the proportion 

A : B : : C : />, 
it gives 

B_D 
A~C 
Then, if we square both members, we have 

and if we cube both members, we havo 

B 3 D 3 

T = C 3 

and then, changing these equalities into a proportion, we have 



for the first, 



A 2 : £ 2 : : C 2 : D . 



and foi the second 

A 2 B* : C 3 : 1) 

Soh. We may verify the proposition by the proportion, 
2 : 4 : : 6 : 12, 
and by squaring each lerm we have, 

4 : 10 : . 36 ■ 144 



BOOK III. 81 



Of Ratios and Proportions 



numbers which are still proportional, and in which the ratio 
is 4. 

If we cube the numbers we have, 



2 3 : 4 3 


:• 3 


12 3 


that is, 8 : f>4 : ■ 


2.6 • 


172 


in which the ratio is 8. 







THEOREM XIV. 

If we have two sets of proportional quantities, the products of 
the corresponding terms will be proportional. 

Let us take the proportions, 
A : B : : C : D which gives 



E : F : : G : H which gives 

Multiplying the equalities together, we have 
BxF DxN 



B_D 
A~C 
F_H 
E~G 



AxE CxG 
md this by Th. II, gives 

AxE : BxF :: CxG : DxH. 

Sch. The proposition may be verified by the followm. 
proportions : 

8 : 12 : : 10 : 15, 
ami 3 : 4 : : 6 : 8 ; 

we sh nil then have 

24 : 48 : : 60 : 120 
whi^h are proportional, the ratio being 9. 



GEO M E T II Y 



BOOK IV 

O * THE MEASUREMENT OF AREAS, AND THD 
PROPORTIONS OF FIGURES. 

DEFINITIONS. 

1 Similar figures, are those which have the angles of tne 
one equal to the angles of the other, each to each, and the 
sides about the equal angles proportional. 

2. Any two sides, or any two angles, which are like placed 
in the two similar figures, are called homologous sides or 
angles. 

3. A polygon which has all its angles equal, each to each, 
and all its sides equal, each to each, is called a regular polygon. 
A regular polygon is both equiangular and equilateral. 

4. If the length of a line be computed in feet, one foot is 
the unit of the line, and is called the linear unit. If the length 
of a line be computed in yards, one yard is the linear unit 

5. If we describe a square on the unit 
of length, such square is called the unit of 
surface. Thus, if the linear unit is one 
foot, one square foot will be the unit of 
surface, or superficial unit. 




BOOK IV. 



S3 



Of Parallelograms 



6. If the linear unit is one yard, one 
square yard will be the unit of surface ; 
and this square yard contains nine square 
feet. 



1 yd. ==3 feet. 





















7. The area of a figure is the measure of its surface. The 
unit of the number which expresses the area, is a square, the 
side of which is the unit of length. 

8. Figures have equal areas, when they contain the same 
measuring unit an equal number of times. 

9. Figuies which have equal areas are called equivalent. 
The term equal, when applied to figures, implies an equality 
in all respects. The term equivalent, implies an equality in 
one respect only : viz. an equality in their areas. The sign 
=0=, denotes equivalency, and is read, is equivalent to. 



THEOREM i. 

Parallclog? ams which, have equal bases and equal altitudes, are 
equivalent. 

Place the base of one parallel- 
ogram on that of the other, so that 
AB shall be the common base of 
the two parallelograms ABCD 
and ABEF. Now, since the par- 
allelograms have the same altitude, their upper bases, DC and 
FE, will fall on the same line FEDC, parallel to AB. Since 
the opposite sides of a parallelogram are equal to each other 
(Bk. I Th. xxiii),.4D is equal to BC. Also, DC and FE are 
each equal to AB : and consequently, they are equal to each 




34 



GEOMETRY 



Of Triangles and Parallelogram j 




other (Ax. 1 ). To each, add ED : 
then will CE be equal to DF. 

But since the line FC cuts the 
two parallels CB and DA, the 
angle BCE will be equal to the 
angle ADF (Bk. I. Th. xiv) : hence, the two triangles ADF 
and BCE have two sides and the included angle of the one 
equal to two sides and the included angle of the other, each 
to each ; consequently, they are equal (Bk. I. Th. iv). 

If then, from the whole space ABCF we take away the tri- 
angle ADF, there will remain the parallellogram ABCD ; but 
if we take away the equal triangle BEC, there will remain the 
parallelogram ABEF : hence, the parallelogram ABEF is 
equivalent to the parallelogram ABCD (Ax. 3). 



Cor. A parallelogram and a 
rectangle, having equal bases and 
equal altitudes, are equivalent. 



THEOREM II 

Triangles which have equal bases and °.qual altitude 
equivalent. 

Place the base of one triangle F D_____E_ 

on that of the other, so that ABC 
and ABD shall be two trian- 
gles, having a common base AB, 
and for their altitude, the distance 
between the two parallels AB, FC : then will the triangle 
ABC be equivalent to the triangle ADB. 

For, through A draw AE parallel to BC, and AF parallel to 
W/). forming the two parallelograms BE and BF Thea 




BOOK IV. 



85 



Of Triangles and Parallelograms 



since these parallelograms have a common base and equal 
altitudes, they will be equivalent (Th. i). 

But the triangle ABC is half the parallelogram BE (Bk. L 
Th. xxiii) ; and A BD is half the equal parallelogram BF . 
hence, the triangle A BC Is equivalent :o the triangle AIM). 




THEOREM 111. 

Lj a triangle and a -parallelogram have equal basts and equal 
altitudes, the triangle will be half the parallelogram. 

Place the base of the triangle on the 
base of the parallelogram, so that AB 
shall be the common base of the tri- 
angle and parallelogram : then will the 
triangle ABE be half the parallelogram 
BD. 

For, draw the diagonal AC. Then, since the altitude of 
the triangle AEB is equal to that of the parallelogram, the 
vertex will be found some where in CD, or in CD produced. 
Now the two triangles ABC and ABE, having the same base 
A D, and equal altitudes, are equivalent (Th. ii). But the tri- 
angle ABC is half the parallelogram BD (Bk. I. Th. xxiii) : 
hence, the triangle ABE is half the parallelogram BD (Ax. IV 

Cor. Hence, if a triangle and a rect- 
angle have equal bases and equal alti- 
tudes, the triangle will be half the 
rectangle. 

For the rectangle would be equiva- 
lent to a parallelogram of the same base 

and altitude (Th. i. Cor.), and since the triangle is half the 
parallelogram, it is also equivalent to half 'he rectangle 




8G 



GEOMETRY. 



Of Rectanglco 



D 



C 11 



A 



B E 



71 



F 



THEOREM IV. 

Rectangles which are described on equal lines are equivalent 

Let BD and FHbe two rectangles, 
having the sides AB, BC, equal to 
the two sides EF, FG, each to 
each: then will the rectangle ABCD, 
described on the lines AB, BC, be 
equivalent to the rectangle EFGH, 
described on the lines EF, FG. 

For, draw the diagonals AC, EG, dividing each parallel- 
ogram into two equal parts. 

Then the two triangles, ABC, EFG, having two sides and 
the included angle of the one equal to two sides and the in- 
cluded angle of the other, each to each, are equal (Bk. 1. 
Th. iv). But these equal triangles are halves of the respective 
rectangles (Th. iii. Cor.) : hence, the rectangles are equal 
(Ax. 7) ; and consequently equivalent. 

Cor. The squares on equal lines are equal. For a square 
is but a rectangle having its sides equal. 



THEOREM V. 



Twc rectangles having equal altitudes are tc each other as their 

bases. 



Let AEFD and EBCF be two 
rectangles having the common alti- 
tude AD ; then will they be to each 
other as the bases AE and EB. 

For, suppose the base A E to be to the base EB, as any two 
numbers, say the numbers 4 and 3 Let AE be then divided 



D _ F 


V 


i J s 1*1 

1 M i I 
i . i ■ i i" 1 


\ 


A E 


B 



BOOK IV. 87 



Of Rectangle 



into four equal parts, and EB into three equal parts, and 
through the points of division draw parallels to AD We 
Bliall thus form seven rectangles, all equivalent to each other 
since they have equal bases and equal altitudes (Th. iv). 

But the rectangle AEFD will contain four of these partial 
rectangles, while the rectangle EBCF will contain three ; 
hence, the rectangle AEFD wil) be to the rectangle EBCF as 
4 to 3 ; that is, as the base AE to the base EB. 

The same reasoning may be applied to any other rect- 
angles whose bases are whole numbers : hence, 

AEFD : EBCF :: AE : EB. 

THEOREM VI. 

Any two rectangles are. to each other as the products of their 
bases and altitudes. 

Let A BCD and AEGF be // I) 

two rectangles : then will 
ABCD : AEGF ■ : ABxAD 
: AFxAE 

For, having placed the two 
rectangles so that BAE and G F 

DAF shall form straight lines, produce the sides CD and GE 
until they meet in H. 

Then, the two rectangles ABCD, AEHD, having the com- 
mon altitude AD, are to each other as their bases AB and 
AE (Th. v)* In like manner, the two rectangles AEHD 
AEGF, having the same altitude AE, are to each other as 
their bases AD and AF. Thus, we have the proportions 
A BCD : AEHD : : AB : AE, 



AEHD : AEGF : : AD : AF. 



8* 



GEOMETRY 



Of Rectangles. 



If, now, we multiply the cor- 
responding terms together, the 
products will be proportional 
(Bk. III. Th. xiv.) ; and the 
common multiplier AEHD may 
be omitted (Bk. III. Th. ix.) : 
hence, we shall have 

ABCD : AEGF : : ABxAD 



H 


d q 


F 




\ 
i 






A B 



G 



AJSXAF. 



Sch. Hence, the product of the base 
by the altitude may be assumed as the 
measure of a rectangle. This product 
will give the number of superficial units 
in the surface : because, for one unit in 
weight, there are as many superficial units 
as there are linear units in the base ; for two units in height, 
twice as many; for three units in height, three times as 
many, &c. 



THEOREM VII. 

The sum of the rectangles contained by one line, arid tht 
several parts of another line any way divided, is equivalent to tlie 
rectangle contained by the two whole lines. 

Let AD be o e line, and AB the other, 
divided into the parts AE, EF, FB : then 
will the rectangles contained by AD and 
AE, AD and EF, AD and FB, be equiv- 
alent to the rectangle A C which is con- 
tained by the lines AD and AB. 

For, through the points E and F draw the lines EG and 
FH, parallel to the line AD : then will the rectangle AG 



D 


G 


H C 


t 






A 


I 


: 


I 


' b 



BOOK I V . 8!) 



Of Areas of Parallelograms. 



be equal to the rectangle of AD x AE ; EH will be equal to 
EGx EF, or to AD x EF; and FC will be equal to FHx FB, 
or to AD x FJ5. 

But the rectangle AC is equal to the sum of the partial 
rectangles : hence, 

ADxAB=G=ADxAE+ADxEF+AD> FB 

THEOREM VIII. 

The area of any parallelogram is equal to the produn of its basr, 
by its altitude. 

Let ABCD be any parallelogram, and 

BE its altitude : then will its area be \~P ~ J 

equal to AB x BE. / 

For, draw AF perpendicular to the !/ 

base AB, and produce CD to F. Then, 



the parallelogram BD and the rectangle EF, having the saint' 
base and altitude are equivalent (Th. i. Cor.). But the arei 
of the rectangle BF is equal to the product of its base AB by 
the altitude AF (Th. vi. Sch.) : hence, the area of the paral 
lelogram is equal to AB x BE. 

Cor. Parallelograms of equal bases are to each other as then 
altitudes ; and if their altitudes are equal, they are to each 
other as their bases. 

For, let B be the common base, and C and D the altitudes 
01' two parallelograms. Then, by the theorem, theii areas are 
to each other, as 

B x C : BxD, 
that is (Bk. III. Th ix), as C : D 

If A and B be their bases, and C their common altitude, 
then they w' 1 ] be to each other as 

A x C : BxC: that is, as A : F 



90 



GEOMETRY 




Areas cf Triang les and Trapezoids. 

THEOREM IX 

The area of a triangle is equal to half &•,„ product of its base by 
its altitude. 

Let ABC be any triangle and CD its 
altitude : then will its area be equal to 
half the product of ABx CD. 

For, through B draw BE parallel to 
AC, and through C draw CE parallel 

to AB : we shall then form the parallelogram AE, having the 
same base and altitude as the triangle ABC. 

But the area of the parallelogram is equal to the product of 
the base AB by its altitude DC ; and since the parallelogram is 
double the triangle (Th. iii), it follows that the area of the tri 
angle is equal to half this product : that is, to half the product 
of ABx CD. 

Cor. Two triangles of the same altitude are to each other 
as their bases ; and two triangles of the same base are to each 
other as their altitudes. And generally, triangles are to each 
other as the products of their bases and altitudes. 



THEOREM X. 



The area of a trapezoid is equal to half the product of its altitud* 
multiplied by the sum of its parallel sides. 

Let ABCD be a trapezoid, CG 
its altitude, and AB, DC its par- 
allel sides : then will its area be 
equal to half the product of 
CGx{AB + DC) 



I) C H ¥ 




\ 

\ 




A C 


? / 


? E 



BOOK IV. 



91 



Of II e c t an £ I 



For, produce AB until BE is equal to DC, and complete 
the rectangle AF ; also, draw BH perpendicular tc AB. 

Then, the rectangle A C will be equivalent to BF, since thev 
have equal bases and equal altitudes (Th iv). The diagonal 
BC will divide the rectangle GH into two equal triangles; 
and hence, the trapezoid A BCD will be equivalent to the 
trapezoid BEFC ; and consequently, the rectangle AF, is 
double the trapezoid ABCD. 

But the rectangle AF is equivalent to the product of 
ADxAE; that is, to CG X [AB-^- DC) ; and consequently 
the trapezoid ABCD is equal to half that product 



THEOREM XI. 



If a line be divided into two parts, the square described on the 
whole line is equivalent to the sum of the squares described an tlie 
two parts, together with twice the rectangle contained by the parti 



Let the line AB be divided into two 
parts at the point E : then wnl the square 
described on AB be equivalent to the two 
squares described on AE and EB, to- 
gether with twice the rectangle contained 
by AE and EB : that is 



D 



H 



F 



AR 



■AE*-{- EB~ + '2AExEB. 



Foi let AC be a square on AB, and A F a sqi are on A E 
and produce the sides EF and GF to H and /. 

Then since EH is equal to AD, being the opposite side of 
a rectangle, it is also equal to AB ; and GI is likewise equaJ 
to AB If thrrefore, from these t-quals we take away EF and 



92 



GEOMETRY. 



Of Rectangle 



D 



H 



G 



GF, there will remain FH equal to FI, 
and each will be equal to HC or IC ; and 
since the angle at F is a right angle, it 
follows that FC is equal to a square de- 
scribed on EB. It also follows, that DF 
and FB are each equal to the rectangle 
of AE into EB. 

But the square ABCD is made up of four parts, viz., the 
square on AE ; the square on EB ; the rectangle DF , and 
the rectangle FB. Hence, the square on AB is equivalent 
to the square on AE plus the square on EB, plus twice the 
rectangle contained by AE and EB. 



B 



Cor. If the line AB be divided into 
two equal parts, the rectangles DF and 
FB would become squares, and the square 
described on the whole line would be 
equivalent to four times the square de- 
scribed on half the line. 



Sch. The property may be expressed in the language uf 
algebra, thus, 

{a + hf=.a+2ab + b? 



THEOREM XII. 



The square described on the hypothenuse of i right angled 
triangle, is equivalent to the sum of the squares described on tfu 
o*her two sides. 



BOOK IV. 



9b 



Of Right Angled Triangles. 




Let BAC be a right an- 
gled triangle, right angled at 
A\ then will the square de- 
scribed on the hypotbenuse 
BC, bo equivalent to tbe two 
squares described on J] A 
and AC. 

Having described the 
squares BC, BL, and AI, 
let fall from A, on the hy- 
potbenuse, the perpendicular p ]g* Q 
AD, and produce it to E\ then draw tbe diagonals AF, CH, 

Now, the angle AJBF is made up of the right angle FBC 
and the angle CBA ; and tbe angle CBIl is made up of the 
right angle ABU and the same angle CBA: hence, tbe angle 
ABF is equal to CJBH. But FB is equal to BC, being sicles 
of the same square; and for a like reason, BA is equal to 
HB. Therefore, the two triangles ABF and CBH, having 
two sides and the included angle of the one equal to two sides 
and tbe included angle of the other, each to eacb, are equal 
(Bk. I. Tb. iv). 

Since tbe angles BAC and BAL are right angles, as 
jlso the angle ABH, it follows that CAL is a straight line 
parallel to BH. (Bk. I. Th. ii. Cor. 3). Hence, the square 
HA and the triangle HBC stand on the same base and be- 
tween tbe same parallels; therefore the triangle is ball* the 
square (Th. iii. Cor.). For a like reason, the triangle A-BF 
is half the rectangle BE. 

But it has already been proved that tbe triangle ABF is 
equal to tbe triangle CJBH : hence, the rectangle BE, which 
is double the former, is equivalent to the square BL, which is 
double the latter (Ax. 6). 



94 



GEOMETRY 



Of Right Angled T r ; a n g 1 e s . 



In the same manner it 
may be proved, that the rect- 
angle DG is equivalent to 
the square CK 

But the two rectangles 
BE, DG, make up the 
square BG : therefore, the 
square BG, described on 
the hypothenuse, is equiva- 
lent to the squares BL and 
CK, described on the other 
two sides. 

Cor. Hence, the square of either side 
of a right angled triangle is equivalent to 
the square of the hypothenuse diminished 
by the square of the other side. That is, 
in the light angled triangle ABC 




AB 4 o 

or £C 2 

Sch. The last theorem 
may be illustrated by de- 
scribing a square on the hy- 
pothenuse BC, equal to 5, 
also on the sides BA, A C, 
respectively equal to 4 and 3 ; 
and observing that the num- 
ber of small squares in the 
large square is equal to the 
number in the two small 
squares 



■■ACT-BCf 



AC'-AB 4 







BOOK IV. 95 



Of Triangle Sides cut Proportionally. 
THEOREM XIII. 

If a line be drawn parallel to the base of a triangle, it will divide 
the other two sides proportionally. 

Let ABC be any triangle, and DE a 
straight line drawn parallel to the base 
BC: then will 

AD : DB : : AE : EC. 




For, draw BE and DC. Then, the 
two triangles BDE and DCE have the 
same base DE, and the same altitude, B C 

since their vertices B and C, lie in the lin< BC parallel to 
DE : hence, they are equivalent (Th. ii). 

Again, the triangles ADE and BDE, ha\e a common ver- 
tex E, and the same altitude ; consequently, they are to each 
other as their bases (Th. ix. Cor.) ; hence, we have 
ADE : BDE : : AD : DB. 

But the triangles ADE and CDE, having a common vertex 

D, are to each other as their bases AE and EC : hence, we 

have 

ADE : CDE : : AE : EC. 

But the triangles BDE and CDE have been proved equiva- 
lent : hence, in the two proportions, the first antecedent and 
consequent in each are equal: therefore, by (Bk. 111. Th- v) 

we have 

AD : BD : : AE : EC. 

. Cor. The sides AB, AC, are also proportional to the parts 
AD, AE, or to BD, CE. 

For, by composition (Bk. III. Th. vii), we have 

AD+BD : BD :: AE+EC : EC. 
Then, by alternation (Bk. 111. Th. iv). 
AB : AC : : BD : EC, hence, also, AB : AC : : AD : AE 



96 



G EOMETRY. 



Proportions of T riant: lei 



THEOREM XIV. 

A line which bisects the vertical angle of a triangle divided 
ihe base into two segments which are proportional to the adjacent 
side. 

Let ACB be a triangk, hav- 
ing the angle C bisected by the 
line CD : then, will 
AD : DB : : AG : CB. 




For, draw BE parallel to 
CD and produce A C to E. 

Then, since CB cuts the two B D A 

parallels CD, EB, the alternate angles BCD and CBE are 
equal (Bk. I. Th. xii) : hence, CBE is equal to angle A CD. 

But, since AE cuts the two parallels CD, BE, the angle 
ACD is equal to GEB (Bk. I. Th. xiv) : consequently, the 
angle CBE is equal to the angle CEB (Ax. I) : hence, the 
«ide CB is equal to CE (Bk. I. Th. vii.) 

Now, in the triangle ABE the line CD is drawn parallel 
to BE: hence, by the last theorem, we have 
AD : DB : : AC i CE, 
and by placing for CE, its equal CB, we have 
AD : DB : : AC : CB. 

THEOREM XV. 

Equiangular triangles have their sides proportional, and are 



Let ABC and DEFbe two equi- 
angular triangles, having the angle 
A equal to the angle D, the angle C 
to the angle F, and the angle B to 
the angle E : then will 

AB : AC : : DE : Z>^ 




BOOK IV. 



97 



Proportions of Triangles. 



For, on the sides of the larger triangle DEF, make Dl 
equal to -AC and DG equal to AB, and join IG. Then thr> 
two triangles ABC and DIG, having two sides and the in- 
cluded angle of the one equal to two sides and the included 
angle of the other, each to each, will be equal (Bk. I Th. iv) 
Hence, the angles / and G are equal to C and B, and conse* 
quently, to the angles F and E : therefore, IG is parallel to 
EF (Bk. I. Th. xiv, Cor. 1 ). 

Now, in the triangle DEF, since IG is parallel to the base, 
we have (Th. xiii). 

DG : Dl : : DE : DF t 



that is, 



AB 



AC 



DL 



DF 



THEOREM XVI. 

Two triangles which have their sides proportional are equian- 
gular and similar. 

Let BAG and EDF be two 
triangles Laving 

BC . EF :: AB : ED, 
and BC : EF : : AC : DF; 
then will they have the corres- 
ponding angles equal, viz.. the angle 

B^E, A = D and C=F. 

For. at the point E make FEG equal to the angle /*, 
and at F make the angle EFG equal to the angle C. Then 
will the angle at G be equal to A, and the two triangles BAC 
and EGF will be equiangukr (Bk. I Th xvii. Cor 1). 

Therefore, by the last theorem, we shall have 

BC : EF : : AB : EG: 
9 




98 



G E0METH1' 



Proportions of T r.i angles. 



AC 



AC 




DF; 



but by hypothesis, 

BC : EF : : AB : DE : 
hence, EG is equal to £D. 

By the last theorem we also 
have 

BC : EF : : 

and by hypothesis, 

BC : EF : : 
hence, FG is equal to DF. 

Therefore, the triangles DEF and EGF, having their three 
sides equal, each to each, are equiangular (Bk. I. Th. viii). 
But, by construction, the triangle EFG is equiangular with 
BAC : hence, the triangles BAC and EDF are equiangular, 
and consequently they are similar. 

Sch. By Theorem XV, it appears that if the corresponding 
angles of two triangles are equal, each to each, the correspond- 
ing sides will be proportional ; and in the last theorem it was 
proved that if the sides are proportional, the corresponding 
angles will be equal. 

Now, these proportions do not hold good in the quadrilate- 
rals. For, in the square and rectangle, the corresponding 
angles are equal, but the sides are not proportional ; and the 
angles of a parallelogram or quadrilateral, may be varied at 
pleasure, without altering the lengths of the sides. 



THEOREM XVII. 



// two triangles have an angle in the one equal to an angle in 
the oihdr, and the sides containing these angles proportional y tke 
two triarchies will he equiangular and similar. 



BOOK 



V . 



99 




Proportion 9 of Tiianglee. 

Let ABC and DEF be two tri- 
angles having the angle A equal to 
the angle D, and 

AS DE : : AC 1 DF ; 

.hen will the two triangles be 
similar. 

For, lay off AG equal to DE, and through G draw GI par 
allel to BC. Then the angle AG I will be equal to the angle 
ABC (Bk. I. Th. xiv) ; and the triangles AGI and ABC will 
be equiangular. Hence, we shall havo 

AB : AG : : AC : AI. 

But, by hypothesis, we have 

AB : DE : : AC : DF, 

and by construction, AG is equal to DE; therefore, AI is 
equal to DF, and consequently, the two triangles AGI and 
DEF are equal in all their parts (Bk. I. Th. it). But the tri- 
angle ABC is similar to AGI, consequently it is similar to 
DEF 



THEOREM XVIII. 

ij from tlie right angle of a right angled triangle, a perpen- 
dicular be let fall on the hypothcnuse, then 

I. The two partial triangles thus formed will be similar to 
each other and to the whole triangle. 

lx. Either side including the right angle will be a mean pro- 
portional between the hypothenuse and the adjacent segment. 

III. The perpendicular will be a mean proportional between the 
segments of the hypothenuse 



100 



GEO M E T H Y 



Proportions of Triangles 




Let ABC be a right angled 
triangle, and AD perpendicular 
to the hypothenuse. 

The two triangles BAC and 
BAD having the common angle 
5, and the right angle BAC equal 
to the right angle at D, will be equiangular (lik. I. Th. xvii 
Cor. 1); and, consequently, similar (Th. xv). For a like 
reason the triangles BAC and CAD are similar. 

Now. from the triangles BA C and BAD, we have 
BC : BA : : BA : BD. 

From the triangles BA C and CAD, we have 
PC : CA : : CA : CD: 
and from the triangles BAD and DAC, we have 
BD : AD :: AD -. DC. 

Cor. If from a point A, in the 
circumference of a circle, AD be 
drawn perpendicular to any diam- 
eter as BC, and the chords AB 
A C be also drawn, then the an- 
gle BA C will be a right angle 
(Bk. II. Th. x): and by the 
theorem we shall have, 

1st The perpendicular AD a mean proportional between 
the segments BD and DC. 

2d Fach chord will be a mean proportional between the 
diameter and the adjacent segment. 

That is, AD*=BDxDC 

AB~=BCxBV 

A~c 2 =r>cxCD 




COOK IV 



11. 



Proportions o f Triangles 



?B 



THEOREM XIX. 

Similar triangles are to each other as the squares described on 
their homologous sides 

Let ABC and DEF be 
two siinilai triangles, and 
A L and DN t\ie squares de- 
scribed on the homologous "TO 
sides AB, DE: then will 
the triangle 

ABC : DEF : : AL : DN. M N 

For, draw CG and FH perpendicular to the bases AB, DE. 
and draw the diagonals B K and EM. 

Then, the similar triangles ABC and DJEF, having their 
homologous sides proportional, we have 
AC : DF : : AB 




and the two ACG, DFB, give 



DE : 



FH; 



AC : DF :: CG 
hence, (Bk. III. Th. v), we have 

AB : DE :: CG : FH, 
or (Bk. III. Th. iv), 

AB : CG : . DE : FH. 

Now, the two triangles ABC and AKB have the common 
base AB ; and the triangles DEF and DEM have the common 
case DE ; and since triangles on equal bases are to each othei 
as their altitudes (Th. ix, Cor.), we have 
he triangle 

ABC : ABK :: CG : AK or AB 
and the triangle, 

DEF : DME : : F!J : DM or DE. 



102 



GEOMETRY. 



Proportions 


of Triangles. 


But we have proved 




CG : AB 


: : FH : DE ; 


hence, ABC : ABK 


: : . DEF : DME, 


or, alternately, 




ABC : DEF 


: ABK : DUE. 




But the squares AL and 
DN being each double of the 
triangles A KB and DME 
liave the sajne ratio ; hence, 

ABC : DEF : : AL : DN 



THEOREM XX. 

Two similar polygons may be divided into an equal number of 

triangles, similar each to each, and similarly placed. 

Let ABCDE and FGHIK be two similar polygons. 

Fiom the angle A draw 
the diagonals AC, AD : J) 

and from the homologous 
angle F, draw FH, FL 

Now, since the poly- 
gons are similar, the ho- 
mologous angles B and G 
will be equal, and the sides about the equal angles propor 
tional (Def. 1): that is, 

AB : BC : : FG : GH. 

Hence, the triangles ABC and FGH have an angle in each 
equal, and the sides about the equal angles proportional . there- 
fore, they are similar (Th. xvii), and consequently, the angle 
ACB is equal to FHG. Taking these from the equal angles 
BCD and GUI, there will remain ACD equal to FHL The 




BOOK IV. 103 



Proportions of Polygons. 

two triangles .4 CD and FHI will then have an angle in each 
equal, and the sides about the equal angles proportional : hence, 
they will be similar. 

In the same manner it may be shown that the triangles 
AED and FKI are similar: and, hence, whatever be the 
number of sides of the polygons, they may be divided into an 
equal number of similar triangles. 

THEOREM XXI. 

Similar polygons are to each other as the squares described on 

their homologous sides. 

Let ABODE and FGNIK, be two similar polygons ; then 
will they be to each other 
as the squares described 
on AB, FG, or any other 
two homologous sides. 

For, let the polygons be 
divided, as in the last the- 
orem, into an equal num- 




F G 



ber of similar triangles. Then, by Theorem XIX, we have 



triangles 








ABC : 


FGN : 


: AJ? . 


FG 2 


ADC 


: FIN : 


: DC? 


: m 2 


ADE 


FIK : 


: DE 2 


: IK 2 



But since the polygons are similar, the ratio of the last ante- 
cedent to its consequent, in each of the proportions, is the 
same : hence, we have (Bk. III. Th. xii). 
ABC+ADC+ADE : FGN+FIN+-FIK : : AB 2 : FG\ 
that is, ABCDE : FGNIK : : AB 2 : FG 2 ; 

Hence, the areas of similar polygons are to each other as 
the squares described on their homologous sides 



104 



GEO M P TRY 



Proportions of Polygons. 




THEOREM XXII. 

If similar polygons are inscribed in circles, tlieir hmnologom 
SvJes, and also their perimeters, will have the s ime t i f ic to each 
Other as the diameters of the circles in which they are inscribed 

Let ABODE, FGNIK, 
be two similar figures, in- 
scribed in the circles whose 
diameters are A L and FM : 
then will each side, AB, 
BC, &c, of the one, be to 
the homologous side FG, GN. &c, of the other, as the 
diameter AL to the diameter ¥M. Also, the perimeter 
AB-\-BC-{- CD &c, will be to the perimeter FG+GN+N1 
&c, as the diameter AL to the dianeter FM 

For, draw the two corresponding diagonals A C, FN, as also 
the lines BL and GM. 

Then, the two triangles ACB and FNG will be similar 
(Th. xx) ; and therefore, the angle A CB is equal to ;he angle 
FNG. But, the angle ACB is equal to the angle ALB, and 
the angle FNG to the angle FMG (Bk. II. Th. ix) : hence, 
the angle ALB is equal to the angle FMG (Ax. J ) ; and since 
ABL and FGM are right angles (Bk. II. Th. x), the two tri- 
angles ALB and FMG will be equiangular (Bk. I. Th. xvii ' 
L'or. 1), and consequently similar (Th. xv). 

Therefore, 

AB : FG : : AL : FM. 
Again, since any two homologous sides are to each other in 
the name ratio as AL to FM, we have (Bk. III. Th xii), 

AB + BC+CD Ac : FG + GN±Nl &c. : : AL : FM. 



BOOK I V . 



105 



Proportions of Polygons. 



THEOREM XXIII. 

Similai polygons inscribed in circles are to each other as the 
squares of the diameters of the circles. 

Let ABODE, FGNIK, 

*ji> Uso polygons inscribed 
in the circles whose diam- 
eters are AL and FM: 
then will the polygon 
ABODE, be to the poly- 
gon FGNIK as the square of AL to the square of FM. 

For, the polygons being similar, are to each other as the 
squares of their like sides (Th. xxi) ; that is, as AB 2 to FG 2 




But, by the last theorem, 

AB : FG 
therefore (Bk III. Th. xiii), 

FG 1 



AL 



AL* 



FM 



FM' 



FGNIK 



AL' 



FM\ 



AB' 
consequently, 

ABODE 

Sch. If any regular polygon, 
ABDEFG,be inscribed in a ciicle, 
and then the arcs AB, BE, &c, be 
bisected, and lines be drawn through 
these points of bisection, a new poly- 
gon will be formed having double the 
number of sides. It is plain that this ^ ~^B 

new polygon Avill differ less from the circle than the first 
polygon, and its sides will lie nearer the circumference than 
the sides ~f the first polygon. 

If now, we suppose the number of sides to be continually 
increased, the length of each side will constantly diminish 




too 



GEOMETRY. 



Proportions of Circles. 



until finally the polygon will become 
equal to the circle, and the perimeter 
will coincide with the circumference. 
When this takes place, the line OH 
drawn perpendicular to one of the 
sides, will become e^ual to the radius 
of the circle. 





THEOREM XXIV. 

The circumferences of circles arc to each other as their diameters 

Let there be two circles 
whose diameters are AL 
and FM: then will their 
circumferences be to each 
other as AL to FM 

For, suppose two similar polygons to be inscribed in the 
circles : their perimeters will be to each other as AL to FM 
(Th. xxii). 

Let us now suppose the arcs which subtend the sides of tho 
polygons to be bisected, and new polygons of double the num- 
ber of sides to be formed : their perimeters will still be to 
each other as AL to FM, and if the number of sides be in- 
creased until the perimeters coincide with the circumference, 
we shall have the circumferences to each other as the diam- 
eters AL and FM. 



THEOREM XXV. 



77*? areas of circles are to each other as the squares of ifair 
diameters. 



BOOK IV 



107 



Area of the Circle. 




Let there be two circles 
whose diameters are AL 
and FM: then will their 
areas be to each other as 
the square of AL to the 
square of FM. 

For, suppose two similar polygons to be inscribed in the 
circles : then will they be to each other as AL 2 to FM* 
(Th xxiii). 

Let us now suppose the number of sides of the polygons to 
be increased, by bisecting the arcs, until their perimeters 
shall coincide with the circumferences of the circles. The 
polygons will then become equal to the circles, and hence, the 
areas of the circles will be to each other as the squares of theii 
diameters. 

Cor. Since the circumferences of circles are to each other 
as their diameters (Th. xxiv), it follows, that the areas which 
are proportional to the squares of the diameters, will also be 
proportional to the squares of the circumferences 

THEOREM XXVI. 

The area of a regular polygon inscribed in a circle, is equal to 
Iwlf the product of the perimeter and the perpendicular let fall 
frwn the centre on one of the sides. 

Let C be the centre of a circle cir- 
cumscribing the regular polygon, and 
CD a perpendicular to one of its sides : 
then will its area be equal to half the 
product of CD by the perimeter. 

For, from C draw radii to the ver- 
tices of the angles, forming as many 




103 



GEOMETRY. 



Area of Circle. 



equal triangles as the polygon has 
sides, in each of which the perpen- 
dicular on the base will be equal to 
CD. Now, the area of one of them, 
as A CB, will be equal to half the pro- 
duct of CD by the base AB ; and the 
same will be true for each of the other 
triangles : hence, the area of the poly- 
gon will be equal to half the product of CD by the perimeter 




THEOREM XXVII. 

Tlie area of a circle is equal to half the product of the radius by 
the circumference. 

Let C be the centre of a circle : 
then will its area be equal to half the 
product of the radius A C by the cir- 
cumference ABE. 

For, inscribe within the circle a 
regular hexagon, and draw CD perpen- 
dicular to one of its sides. Then, 
the area of the polygon will be equal to half the product oi 
3D multiplied by the perimeter (Th. xxvi). 

Let us now suppose the number of sides of the polygon to 
oe increased, until the perimeter shall coincide with the cir- 
cumference ; the polygon will then become equal to the chela 
and the perpendicular CD to the radius CA. Hence, the area 
of the circle will be equal to half the product of the radius by 
the circumference. 




BOOK IV. 109 



Pr o d 1 e m s 



PROBLEMS 



RELATIXG TO THE FOURTH BOOK. 




PROBLEM I. 

To divide a line into any proposed number of equal parti 

Let AB be the line, and let it be 
required to divide it into four equal 
parts. 

Draw any other line, A C, forming 
an angle with AB, and take any dis- 
tance, as AD, and lay it off four times on A C. Join C and B 
and through the points D, E, and F, draw parallels to CB 
These parallels to BC will divide the line AB into parts pro 
portional to the divisions on A C (Th. xiii) : that is, into equal 
parts. 

PROBLEM II. 
To find a third proportional to two given lines. 
Let A and B be the given lines. 

Make AB equal to A, and draw p ~~_ 

C 
A C, making an angle with it. On "^"V 

A C lay oflf AC equal to B, and join \j\ 

HC . then lay off AD, also equal to 

]) and through D draw DE parallel to BC : ihen will AE 

be the third proportional sought 

For, since DE is parallel to BC, we have (Th. xiii) 

AB : AC : : AD or AC : AE; 

♦herefore, AE is the third proportional sought 
10 



B 



I J GEOMETRY 



Problems 



PROBLEM III. 
To find a fourth proportional to the lines A, B, and C. 
Place two of the lines forming an 



A 

angle with each other at A ; that is, # 

make AB equal to A, and AC equal C K^\ 

B ; also, lay off AD equal to C. ^s^ \ 



Then join BC, and through D draw A V B 

DE parallel to BC, and AE will be the fourth proportional 
sought. 

For, since DE is parallel to BC, we have 

AB : AC n AD : AE; 
therefore, AE is the fourth proportional sought. 

PROBLEM IV. 

To find a mean proportional between two given lines, A and b 

Make AB equal to A, and 
BC equal to B: on AC de- 
scribe a semicircle. Through 
B draw BE perpendicular to 
A C, and it will be the mean proportional sought (Th. xviii. Cor) 

PROBLEM V. 

To make a square which shall be equivalent to the sum of twe 
given squares. 

Let A and B be the sides of the 
given squares. 

Draw an indefinite line AB, and 
make AB equal to A. At B draw 
BC perpendicular to AB, and make 
BC equal to B : then draw A C and the square described on 
AC will be equivalent to the squares on A and B (Th. xii). 





BOOK IV. Ill 



Problems 



PROBLEM VI. 

To make a square vuhich shall be equivalent to the difference be 
tween two given squares. 

Let A and B be the sides of 



, a 



'B 



»lie given squares. 

Draw an indefinite line, and 

make CB equal to A, and CD ' q — ~jy 

equal to B. At D draw DE 
perpendicular to CB, and with C as a centre, and CB as a 
radius, describe a semicircle meeting DE in E, and join CE: 
then will the square described on ED be equal to the differ- 
ence between the given squares. 

For, CE is equal to CB, that is, equal to A, and CD is 
equal to B : and by (Th. xii. Cor.), 

ED l =CE 2 -CD l . 

PROBLEM VII. 

To make a triangle which shall be equivalent to a given quad- 
rilateral. 

Let A BCD be the given quadri- 
ateral. 

Draw the diagonal A C, and through 
D draw DE parallel to AC, meeting 
BA produced at E. Join EC: then will the triangle CEB 
be equivalent to the quadrilateral BD. 

For, the two triangles ACE and ADC, having the same base 
A C, and the vertices of the angles D and E in the same line 
DE parallel to AC, are equivalent (Th. ix). If to each, we 
add ACB, we shall then have the triangle ECB equivalent to 
the quadrilateral BD (Ax. 2). 




E A B 



112 



CEO I\l E T K Y 



Problems 



PROBLEM VIII. 

To make a triangle which shall be equivalent to a given polygon 

Let ABODE be the polygon. 

Draw the diagonals AD, BD. 
Produce AB in both directions, 
and through C and E draw CG 
and EF, respectively parallel to 
AD and BD : then join FD and 
DG, and the triangle FDG will be equivalent to the polygor, 
ABODE. 

For, the triangle AED is equivalent to the triangle AFD 
and DBC to DBG (Th. ii); and by adding ADB to the 
equals, we shall have the triangle FDG equivalent to the 
polygon ABODE. 




PROBLEM IX. 

To make a rectangle that shall be equivalent to a given triangle. 

Let ABC be the given triangle. 

Bisect the base AB at D, and draw 
DH perpendicular to AB. Through C, 
the vertex of the triangle, draw CHG 
parallel to AB, and draw BG perpen- 
dicular to it : then will the rectangle 
DG be equivalent to the triangle ABC. 

For, the triangle would be half a rectangle having the same 
base and altitude : hence, it is equivalent to DG, whose base 
ia the half of AB, and altitude equal to that of the triangle. 




BOOK IV 



113 



Appendix 




PROBLEM X. 
To inscribe a circle in a regular polygon. 

Bisect any two sides of the polygon 
by the perpendiculars GO, FO, and 
with their point of intersection O, as a 
centre, and OGas a radius describe 
the circumference of a circle — this 
circle will touch all the sides of the 
polygon. 

For, draw OA. Then in the two right angled triangles OA G 
and OAF, the side AO is common, and A G is equal to if, 
since each is half of one of the equal sides of the polygon : 
hence, OG is equal to OF(Bk. I.Th. xix). In the same man- 
ner it may be shown that OH, OK and OL are all equal to 
each other : hence, a circle described with the centre O and 
radius OF will be inscribed in the polygon. 

C.r. Hence, also the lines OA, ON &c, drawn to the 
angles of the polygon are equal. 



APPENDIX 



OF THE REGULAR POLYGONS. 

1. In a regular polygon the angles are all equal to each 
other (Def. 3). If then, the sum of the inward angles of a 
regular polygon be divided by the number of angles, the quo- 
tient will be the value of one of the angles. 

But the sum of the inward angles is equal to twice as many 
right angles, wanting four, as the polygon has sides, and we 
shall find the value in degrees by simply placing 90° for the 
right angle. 



114 GEOMETRY. 

Appendix. ^^ 

2. Thus, for the sum of all the angles of an equilateral 
'riangle, we have 

6x90°-4x90 o = 540 o -360 o = 180° 
and foi each angle 

180°-f-3 = 60°: 
Hence, each angle of an equilateral triangle, is equal to GO 
degrees. 

3. For the sum of all the angles of a square, we have 

8x90°-4x90 o z=720 o -360 o = 360°, 
nnd for each of the angles 

360° ^4 = 90° 

4. For the sum of all the angles of a regular pentagon, wc 
have 

10 x 90° -4 X 90° = 900° -360° = 540°, 
and for each angle 

540° ^-5= 108°. 

5. For the sum of all the angles of a regular hexagon, we 
have 

12 x 90° -4 x 90° = 1080° -360° = 720°, 
and of each angle 

720°^ 6 = 120°. 

6. For the sum of the angles of a regular heptagon, we 
have 

14x90°-4x90° = 1260° — 360° = 900°: 
and for one of the angles 

900° h- 7 = 128° 34'+. 

7. For the sum of the angles of a regular octagon, we ha\o 

16x90°-4x90 o = 1440 o -360°=rl080 o : 
and for each angle 

l080°--r8=13S p 



BOOK IV 



115 



Regular Polygons. 



8. Since the sum of the angles about any point is equal tc 
four right angles (Bk. I. TL ii. Cor. 3), it may be observed thai 
there are only three kinds of regular polygons, which can be 
ai ranged around any point, as C, so as exactly to fill up the 
space. These are, 



First. — Six equilateral triangles, in 
which each angle about C is equal to 
■60°, and their sum to 

60° x 6 = 360. 




Second.- Four squares, in which 
each angle is equal to 90°, and their 
si un to 

90° x 4 = 360° 



n 



Third. — Three hexagons, in 
-.vhich each angle is equai to 
120, and the sum of the three 
to 

120° X 3 = 360°. 




GEOMETRY, 



BOOK V. 

OF PLANES AND .THEIR ANGLES. 
DEFINITIONS. 

1 . A straight line is perpendicular to a plane, when it is per 
pendicular to every straight line of the plane which it meets 
The point at which the perpendicular meets the plane, is 
called the foot of the perpendicular. 

2. If a straight line is perpendicular to a plane, the piano 
is also said to be perpendicular to the line. 

3. A line is parallel to a plane when it will not meet that 
plane, to whatever distance both may be produced. Con- 
versely, the plane is then parallel to the line. 

4. Two planes are parallel to each other, when they will 
not meet, to whatever distance both are produced. 

5. If two planes are not parallel, they intersect each other 
in a line that is common to both planes : such line is called 
their common intersection. 

6. The space included between two planes is called a 
diedral angle : the planes are the faces of the angle, and 
their intersection the edge. A diedral angle is measured by 
two lines, one in each plane, and both perpendicular to the 
common intersection at the same point. 

This angle may be acute, obtuse, or a right angle. When 
it is a right angle, the planes are said to be perpendicular tc 
each other. 



BOOK 



117 



Of Planes. 



Z> 



B 



E 



J 



Let AB be a plane coinciding with H 

the j lane of the paper, and ECF a 
plane intersecting it in the line FH. 
Now, if from any point of the common 
intersection as C, we draw CD in the 
plane AB, and CE in the plane ECF, 
and both perpendicular to CF at C, 
then will the angle DCE measure the inclination between 
the two planes. 

It should be remembered that the line £C is direct Iv ov«i 
the line CD. 

7. A polyedral angle is the angular 
space included between several planes 
meeting at the same point. 

Thus, the polyedral angle S is formed 
by the meeting of the planes ASB, 
BSC, CSD, DSA. 

8. The angle formed by three planes 
is called a triedral angle. 




THEOREM I. 

Two straight lines which intersect each other, lie ni tliv. saw. 
plane, and determine its position. 

Let A B and AC be two straight lines 
which intersect each other at A. 

Through AB conceive a plane to be 
pft3sed, and iet this plane be turned 
around AB until it embraces the point 
C : the plane will then contain the two 

lines AB, AC, and if it be turned either way it will depart 
from the point C, and consequently from the line A C. Hence, 




118 



GEOMETRY. 



Of Planes. 



the position of the plane is determined 
by the single condition of containing 
the two straight lines AB, A C. 

Cor. 1. A triangle ABC, or three 
points A, B, C, not in a straight line, 
determine the position of a plane. 

Cor. 2. Hence, also, two parallels 
AB, CD determine the position of a 
plane. For drawing EF, we see that 
the plane of the two straight lines AE, 
EF is that of the parallels AB, CD. 




K 



C /F 



THEOREM II. 

.1 •perpendicular ts the shortest line which can be drawn from a 
point to a plane. 

Let A be a point above the plane 
DE, and AB a. line drawn perpen- 
dicular to the plane : then will A B be 
shorter than any oblique line A C. 

For, tlirough B, the foot of the per- 
pendicular, draw BC to the point 
where the oblique line A C meets the 
plane. 

Now, since A B is perpendicular to 
the plane, the angle ABC will be a 

right angle (Def. 1.), and consequently less than the angle C: 
therefore, AB, opposite the angle C, will be less than AC 
opposite the angle B (Bk. I. Th. xi). 




BOOK V 



11G 



Of Planes. 



Cor It is evident that if several lines be drawn from the 
point A to the plane, that those which are nearest the perpen- 
dicular AB, will be less than those more remote. 

Sch. The distance from a point to a plane is measured on 
die perpendicular : hence, when the distance only is named, 
the shortest distance is always understood. 




THEOREM III. 

The common intersection of two planes is a straight line 

Let the two planes AB, CD, cut 
each other. Join any two points E 
and F, in the common intersection, i.^ 
by the straight line EF. This line 
will lie wholly in the plane AB, and 
also wholly in the plane CD (Bk. I. 
Def. 7) ; therefore, it will be in both 
planes at once, and consequently, is 
their common intersection. 



.JB 




THEOREM IV. 

A straight line which is perpendicular to two straight lines at 
their point of intersection, will be perpendicular to the plane of 
those lines. 

Let the line PA be perpen- 
dicular to the two lines AD, 
AB : then will it be perpendic- 
ular to the plane BC which con- 
tains them. 

For, if AP is not perpendicular 
to the plane BC, suppose a plane 




120 



GEOMETRY 



Of Plan 



to be drawn through A, that shall 
be peqiendicular to AP 

Now. every line drawn through 
,4, and perpendicular to A P. 
n'ill be a line of this last plane 
(l)ef. 1): hence, this last plane 
will contain the lines AB, AD, 
and consequently, a line winch is perpendicular to two lines 
at the point of intersection, will be perpendicular to the plane 
of those lines. 



D 


C 


p 


A 



B 



THEOREM V. 

if two straight lines are perpendicular to the same plane they 
will be parallel to each other. 

Let the two lines AB, CD, be 
perpendicular to the plane EF : 
then will they be parallel to each 
other 

For, join B and D, the points 
in which the lines meet the 
plane EF 

Then, because the lines AB, CD, are perpendicular to the 
plane EF, they will be perpendicular to the line BD (Def. 1). 
Now, if BA and DC are not parallel, they will meet at some 
point as : then, the triangle OBD would have two right 
angles, which ie impossible (Bk. I. Th. xvii. Cor. 4). 

Cor. If two lines are parallel, and one of them is perpen- 
dicular to a plane, the other will also be perpendicular to the 
same plane. 



A C 

A 1 » 




l\ 




I 


>' / 


> 



KOOK V 



121 



Of Planea. 




THEOREM VI. 

If two planes intersect each other at right angles, and a line 
bo drawn in one plane perpendicular to the common intersection t 
this line will be perpendicular to the other plane. 

Let the plane FE be perpen- 
dicular to MN, and AP be drawn 
m the plane FE, and perpen- 
dicular to the common intersec- 
tion DE: then will AP be per- 
pendicular to the plane MN. 

For, in the plane MN draw 
CP perpendicular to the common 
intersection DE. Then, because the planes MN and FE arc- 
perpendicular to each other, the angle APC, which measures 
their inclination, will be a right angle (Def. 6). Therefore, 
the line AP is perpendicular to the two straight lines PC and 
PD ; hence, it is perpendicular to their plane MN (Th. iv). 

THEOREM VII. 

If one p'anc intersect another plane, the sum of the angles on 
he same side will be equal to two right angles. 

Let the plane GEF intersect 
the plane A B in the line FE : 
then will the sum of the two 
angles on the SMne side be equal 
to two right angles. 

! or, from any point, as E, in 
llio common intersection, draw 
the lines EG and DEC, one in each plane, and botn perpen- 
dicular to the common intersection at E. Then, the line G U 

makes, with the line DEC, two angles, which together are 
11 




12'J 



GEOMETRY. 



Of Planca 






equal to two right angles (Bk I. 
Th. ii): but these angles measure 
the inclination of the planes ; there- 
fure, the sum of the angles on the 
same side, which two planes make 
with each other, is equal to two 
right angles. 

Cor. In like manner it may be demonstrated, that planes 
which intersect each other have their vertical or opposite 
angles equal. 

THEOREM VIII. 

Two planes whch are perpendicular to the same straight line we 
parallel to each other. 

Let the planes MN and PQ 
be perpendicular to the line AB: q 
then will they be parallel. \. 

For, if they can meet any 
where, let O be one of their 
their common points, and draw 

OB, in the plane PQ, and OA, \ __\ 

in the plane MN. 

Now, since AB is perpendicular to both planes, it will 
be perpendicular to OB and OA (Def. 1) : hence, the triangle 
OAB will have two right angles, which is impossible (Bk. I. 
Th. xvii. Cor. 4) ; therefore, the pianes can have no point, ds 
0, in common, and consequently, they are parallel (Def 4). 



M 



D 



N 



THEOREM IX. 



If a plane cuts two parallel planes, the lines of intersection wilJ 
be parallel 



13 O O K V 



123 



Of Plane 



Let the parallel planes MIS and 
PA be intersected by the plane 
EH: then will the lines of inter- 
section EF, GH, be parallel. 

For, if the lines EF, GH, were 
not parallel, they would meet each 
other if sufficiently produced, since 
they lie in the same plane. If this 
were so, the planes MN, PA , would 
meet each other, and, consequently, could not be parallel; 
which would be contrary to the supposition. 




THEOREM X. 

Ij two lines are parallel to a third line, though not in the sama 
plane with it, they will be parallel to each other. 

Let the lines AB and CD be each 
parallel to the third line EF, though 
not in the same plane with it : then 
will they be parallel to each other. 

Foi since EF and CD are parallel, 
they will lie in the same plane FC 
(Th. i. Cor. 2), and AD, EF will also 
lie in the plane EB. 

At any point, G, in the line EF, let GI ami GH be drawn 
in the planes FC, BE, and each perpendicular to FE at G 

Then, since the line EF is perpendicular to the lines GH 
GI, it will be perpendicular to the plane HGI (Th. iv). And 
since FE is perpendicular to the plane HGI, its parallels 
AB and DC will also be perpendicular to the same plane 
(Th. v). Hence, since the two lines AB, CD, are both per* 
pellicular to the plane HGI, the} will be parallel to each other 




12 1 



G E () M E T R Y 



Of Planes 



THEOREM XJ. 

If two angles, not situated in the same plane, have their side j 
parallel and lying in the same direction, the angles will br> 
equal. 

Let the angles ACE and BDF 
have the sides AC parallel to BD, 
and CE to DF: then will the angle 
ACE be equal to the angle BDF. 

For, make AC equal to BD, and 
CE equal to DF, and join A B, CD, 
iwA EF ; also, draw AE, BF. 

Now since AC is equal and par- 
allel to BD, the figure AD will be a 
parallelogram (Bk. 1. Th. xxv); there- 
fore, AB is equal and parallel to CD. 

Again, since CE is equal and parallel to DF, CF will be 
a parallelogram, and EF will be equal and parallel to CD. 
Then, since AB and EF are both parallel to CD, they will 
be parallel to each other (Th. x) ; and since they are each 
equal to CD, they will be equal to each other. Hence, the 
figure BAEF is a parallelogram (Bk. 1. Th. xxv), and conse- 
quently, AE is equal to BF. Hence, the two triangles ACE 
and BDF have the th^ee sides of the one equal to the three 
sides of the other, each to each, and therefore the angle AC'E 
is equal to the angle BDF (Bk. I. Th. viii). 




THEOREM XII. 



If tu?o planes are parallel, a straight line vihich is perpendicular 
to the one will also be perpendicular to the oilier. 



BOOK v 



123 



Of Plane 



M 



Let MN and PQ be two par- 
allel planes, and let AB be per- 
pendicular to MN : then will it 
be perpendicular to PQ. 

For, draw any line, BC, in the 
plane PQ, and through the lines 
AB, BC, suppose the plane 
ABC to be drawn, intersecting 
the plane MN m the line AD : then, the intersection AD will 
be parallel to BC (Th. ix). But since AB is perpendicular 
to the plane NM, it. will be perpendicular to the straight line 
AD, and consequently, to its parallel BC (Bk. I. Th. xii. Cor.) 

In like manner, AB might be proved perpendicular to any 
other line of the plane PQ, which should pass through B ; 



\ *\ A 


r> 


N 


\ 


\ 





hence, it is perpendicular to the plane (\)v\\ 1). 

Cor. It from any point as H, 
any oblique linos, as HEF, HDC, 
be drawn, the parallel planes will 
cut these lines proportionally. 

For, draw II AB perpendicular 
to the plane MN : then, by the 
theorem, it will also be perpendi- 
cular to PQ. Then draw AD, AE, 
BC, BE. Now, since AE, BE, 
are the intersections of the plane 
FIIB, with the two parallel planes MN, PQ, they are paral- 
lel (Th ix.); and so also are AD, BC. 




Then, HA 


ITB 


: HE 


HE, 


and HA 


IIP 


HD 


HC, 


hence, HE : 


HE : 


: HD 


HC 


11* 









GEOMETRY. 



BOOK VI 



OF SOLIDS. 



DEFINITIONS 



1. Ever}' solid bounded by planes is called a polyadnm. 

2. The planes which bound a polyedron are called facsa. 
The straight lines in which the faces intersect each other, 
are called the edges of the polyedron, and the points at which 
the edges intersect, are called the vertices of the angles, or 
vertices of the polyedron. 

3. Two polyedrons are similar, when they are contained by 
the same number of similar planes, and have their polyedral 
angles equal, each to each. 

4. A prism is a solid, whose ends 
arc equal polygons, and whose side 
faces are parallelograms. 

Thus, the prism whose lower base 
is the pentagon ABCDE, terminates 
in an equal and parallel pentagon 
FGHIK, which is called the upper 
base. The side faces of the prism 
are the parallelograms DH, DK, EF, 
A G, and BH. These are called the convex, or lateral surface 
of the nrism 




ROOK V 



.27 



Of the Prism 



5. The altitude of a prism is the distance between its upper 
and lower bases : that is, it is a line drawn from a point of the 
upper base, perpendicular, to the lower base 



6, A right prism is one in which 
the edges AF, BC, EK, HC, and 
DI, are perpendicular to the bases. 
In the right prism, either of the per- 
pendicular edges is equal to the 
altitude. In the oblique prism the 
altitude is less than the edge. 



K 



'£'--. 



H 



/ 



D 



B C 



7'. A prism whose base is a triangle, is called a triangular 
prism ; if the base is a quadrangle, it is called a quadrangular 
prism ; if a pentagon, a pentagonal prism ; if a hexagon a 
hexagonal prism ; <&c. 



8. A prism whose base is a parallelo- 
gram, and all of whose faces are also 
parallelograms, is called a parallelopipe- 
don. If all the faces are rectangles, it is 
called a rectangular parallelopipedon. 



9. If the faces of the rectangular par- 
allelopipedon are squares, the solid is 
called a cube: hence, the cube is a prism 
bounded by six equal squares 




128 



GEOMETRY. 



Of the Pyramid 



10. A pyramid is a solid, formed by 
several triangles united at the same 
point S, and terminating in the differ- 
ent sides of a polygon ABCDE. 

The polygon ABCDE, is called the 
lise of the pyramid ; the point *S, is 
called the vertex, and the triangles 
ASB, BSC, CSD, DSE, and ESA. 
form its lateral, or convex surface. 




1 1 . A pyramid whose base is a triangle, is palled a titan- 
gular pyramid ; if the base is a quadrangle, it is called a 
quadrangular pyramid ; if a pentagon, it is called a petagonal 
pyramid; if the base is a hexagon, it is called a hexagonal 
pyramid; &c. 



12. The altitude of a pyramid, is the 
perpendicular let fall from the vertex, 
upon the plane of the base. Thus, 
SO is the altitude of the pyramid 
* —ABCDE. 




13. When the base of a pyramid is a. regular polygon, and 
the perpendicular SO passes through the middle point of the 
base, the pyramid is called a right pyramid, and the line 
SO is called the axis 



BOOK VI. 



J2U 



Pyramid and Cylinder. 



14. The slant height of a right 
pyramid, is a line drawn from the ver- 
tex, perpendicular to one of the sides 
of the polygon which forms its base. 
Thus. SF is the slant height of the 
pyramid S— ABODE. 



15. If from the pyramid S—ABCDE 
the pyramid S — abede be cut off by a 
plane parallel to the base, the remain- 
ing solid, below the plane, is called 
the frustum of a pyramid. 

The altitude of a frustum is the per- 
pendicular distance between the upper 
and lower planes. 




16. A Cylinder is a solid, described by 
the revolution of a rectangle, AEFD, 
about a fixed side, EF. 

As the rectangle AEFD, turns around 
the side EF, like a door upon its hinges, 
the lines AE and FD describe circles, 
and the line AD describes the convex sur- 
face of the cylinder. 

The circle described by the line AE, is called the 
base of the cylinder, and the circle described by DF, is 
\he upper bas-e.. 




lower 
callod 



130 



GEOMETRY 



Of the Cy Under 



The immovable line EF is called the axis of the cylinder 
A cylinder, therefore, is a round body with circular ends 



17. If a plane be passed through the 
axis of a cylinder, it will intersect the cylin- 
der in a rectangle, P#, which is double 
the revolving rectangle DE. 




^F 



18. ff a cylinder be cut by a plane par- 
allel to the base, the section will be a cir- 
cle equal to the base. For, while the 
side FC y of the rectangle MC, describes 
the lower base, the equal side MP, will 
describe the circle MLKN, equal to the 
lower base. 




19 If a polygon be inscribed in the 
lower base of a cylinder, and a corres- 
ponding polygon be inscribed in the upper 
base, and their vertices be joined by 
straight lines, the prism thus formed is 
said to be inscribed in the cylinder. 




BOOK VI. 



L31 



Of the Cone 




20. A cone is a solid, described by 
the revolution of a right angled triangle, 
ABC, about one of its sides, CB. 

The circle described by the revolving 
side, AB, is called the base of the cone. 

The hypothenuse, AC, is called the 
slant height of the cone, and the surface 
described by it, is called the convex 
surface of the cone. 

The side of the triangle, CB, which remains fixed, is called 
the axis, or altitude of the cone, and the point C, the vertex 
of the cone. 

21. If a cone be cut by a plane par- 
allel to the base, the section will be a 
circle. For, while in the revolution of 
the right angled triangle SAC, the line 
CA describes the base of the cone, its 
parallel FG will describe a circle 
FKHI, parallel to the base. If from 
the cone S— CDB,the cone S—FKH 
be taken away, the remaining part is 
called the frustum of the cone 




22. If a polygon be inscribed 
in the base of a cone, and straight 
lines be drawn from its vertices 
to the vertex of the cone, the pyra- 
mid thus formed is said to be in- 
scribed in the cone. Thus, the 
pyramid S — ABCD is inscribed in 
the cone 




132 



GEOMETRY 



Of the Sphere, 



23. Two cylinders are similar, when the diameters of their 
bases are proportional to their altitudes. 

24. Two cones are also similar, when the diameters of theii 
bases are proportional to their altitudes. 

25. A sphere is a solid terminated by a curved surface, ali 
the points of which are equally distant from a certain point 
within called the centre. 



26. The sphere may be described 
by revolving a semicircle, ABD, 
about the diameter AD. The plane 
will describe the solid sphere, and 
the semicircumference ABD will 
describe the surface. 




27. The radius of a sphere is a 
line drawn from the centre to any 
point of the circumference. Thus, 
CA is a radius. 




28. The diameter of a sphere is 
a line passing through the centre, 
and terminated by the circumfer 
once. Thus. AD is a diameter 




BOOK VI. 



L33 



Of the Sphere 



29. All diameters of a sphere are equal to each other; and 
each is double a radius. 

30. The axis of a sphere is any line about which it re- 
volves ; and the points at which the axis meets the surface, 
ire called the poles. 



31. A plane is tangent to a sphere 
when it has but one point in com- 
mon with it. Thus, AB is a tan- 
gent plane, touching the sphere at B< 




32. A zone is a portion of the sur- 
face of a sphere, included between 
two parallel planes which form its 
bases. Thus, the part of the surface 
included between the planes AE 
and DF is a zone. The bases of 
this zone are the two circles whose 
diameters are AE and DF. 




33. One of the planes which 
bound a zone may become tangent 
to the sphere ; in which case the 
zone will have but one base. Thus, 
if one plane be tangent to the sphere 
at A, and another plane cut it in the 
circle DF, the zone included be- 
tween them, will have but one base. 
12 




134 



GEOMETRY 



Of the P i ; s m 



34. A spherical segment is a portion of the solid sphere in- 
cluded between two parallel planes. These parallel planes 
are its bases. If one of the planes is tangent to the sphere, 
the segment will have but one base. 

35. The altitude of a zone or segment, is the distance be 
Kveen the parallel planes which form its bases 



t 



THEOREM I. 

The convex surface of a right prism is equal to the perimeter of 
its base multiplied by its altitude. 

Let ABODE— K be a right 

A 
prism : then will its convex surface 

be equal to 

{AB-) BC+CD + DE-{-EA)xAF. 

For, the convex surface is equal 
to the sum of the rectangles AG> 
BH, CI, DK, and EF, which com- 
pose it ; and the area of each rectan- 
gle is equal to the product of its base B U 
by its altitude. But the altitude of each rectangle is equal to 
the altitude of the prism : hence, their areas, that is, the con- 
vex surface of the prism, is equal to 

(AB + BC + CD+DE + EA) x A F; 

that is, equal to the perimeter of the base of the prism multi 
plied by its altitude. 



E'"< 



D 



THEOREM II. 

The convex surface of a cylinder is equal to the circumference of 

its base multiplied by its altitude 



BOOK VI. 



135 



Of the Pr 




Let DB be a cylinder, and AB the 
diameter of its base : the convex sur- 
face will then be equal to the altitude 
4 /) multi] lied by the circumference 
of the base. 

Tor, suppose a regular prism to be 
inscribed within the cylinder. Then, 
the convex surface of the prism will be 
equal to the perimeter of the base mul- 
tiplied by the altitude (Th. i). But the altitude of the prism 
is the same as that of the cylinder ; and if we suppose the 
sides of the polygon, which forms the base of the prism, to 
be indefinitely increased, the polygon will become the circle 
(Bk. IV.Th.xxiii. Sch.), in which case, its perimeter will become 
the circumference, and the prism will coincide with the cylinder. 
But its convex surface is still equal to the perimeter of its base 
multiplied by its altitude: hence, the convex surface of a cylin- 
der is equal to the circumference of its base multiplied by its al- 
titude. 

THEOREM III. 

In every prism the sections formed by planes pirallcl to the bast 

are equal polygons. 

Let A G be any prism, and IL a sec- 
tion made by a plane parallel to the 
base AC: then will the polygon IL 
be equal to A C. 

For, the two planes A C, IL, being 
parallel, the lines AB, IK, in which 
they intersect the plane AF, will also 
be parallel (Bk. V. Th. ix). For a 
like reason, BC and KL will be par- 





K 


\ 


G 


/ 






\ 
\ 


A 


\ 

\ 


K 

I) 


i 


, I 


3 


( 


> 



136 



GEOMETRY 



Of the Pyramid 



allcl; also, CD will be parallel to LM, 
and AD to IM. 

But, since AI and BK are parallel, 
the figure AK is a parallelogram : 
hence AB is equal to IK (Bk. 1. 
Th. xxiii). In the same way it may be 
shown that BC is equal to KL, CD to 
LM, and AD to IM. 

But, since the sides of the polygon 
AC are respectively parallel to the 
sides of the polygon IL, it follows that their corresponding 
angles are equal (Bk. V. Th. xi), viz., the angle A to the angle 
/, the angle B to K, the angle C to L, and the angle M to D ; 
hence, the polygon IL is equal to AC. 

Sch. It was shown in Definition 18, that the section oi a 
cylinder, by a plane parallel to the base, is a circle equal to 
the base. 



\ 






\ 
\ 


\ 




M 




A' 






\ 


D \ 



B 



THEOREM IV. 

If a pyramid be cut by a "plane parallel to the base, 

I. The edges and altitude will be divided proportionally. 

II. The section ioill be a. polygon similar to the base. 
Let the pyramid S—ABCDE, of 

which SO is the altitude, be cut by the 
plane abede parallel to the base : then 
will, 

Sa : SA : : Sb : SB, 

and the same for the other edges ; and 
the polygon abede will be similar to the 
base ABCDE. 

First, Since the planes A BC and ah*- 




B O O R V 1 . 137 



Of the Pyramid 



are parallel, their intersections, AB, ab, by the plane SA B, 
will also be parallel (Bk. V. Th. ix) ; hence, the triangles 
SAB. sab, are similar, and we have 

SA : Sa : : SB : Sb ; 
fax a similar reason, we have 

SB : Sb '. : SC : Sc ; 

end the same for the other edges ■ hence, the edges SA, SB % 
SC, &c, are cut proportionally at the points a, b, c, &c. 
The altitude SO is likewise cut proportionally at the point 
The altitude SO is likewise cut in the same proportion at 
the point o ; for, since BO is parallel to bo, we have 

SO : So : : SB : Sb. 

Secondly. Since ab is parallel to AB, be to BC, cd to CD 
Sic. ; the angle abc is equal to ABC, the angle bed to BCD 
and so on (Bk. V. Th. xi). 

Also, by reason of the similar triangles, SAB, Sab, we have 

AB : ab : SB : Sb, 

and by reason of the similar triangles SBC, Sbc, we have 
SB : Sb : : BC : be; 

hence (Bk III. Th. v), 

AB : ab : : BC : be; 

and for a similar reason, we also have 

BC : be : : CD : cd, Sic. 

Hence, tne polygons ABCDE, abede, having their angles 
respectively equal, and their homologous sides proportional 
are similar. 

12* 



L3S 



GEOMETRY. 



Of the Pyramid. 



THEOREM V. 

If two pyramids, having equal altitudes and their bases in the, 
same plane, be intersected by planes parallel to the plane of lh& 
bases, the sections in each pyramid v:ill fi proportional to the bases 

Let S—ABCDE, and 
S — XYZ, be two pyra- 
mids, having a common 
vertex, and their bases sit- 
uated in the same plane. 
If these pyramids are cut 
by a plane parallel to the 
plane of their bases, giv- 
ing the sections abede, A ^^~~^1 
xyz, then will the sections Y 
abede, xyz, be to each other as the bases ABCDE, XYZ. 

For, the polygons ABCDE, abede, being similar, their sur* 
faces are as the squares of the homologous sides AB, ab ; 

but AB : ab : : ■ SA : Sa: 




abede 



SA' 



Sa' 



hence, ABCDE 
For the same reason, 

XYZ : xyz : : SX* : «?. 
But since abc and xyz are in one plane, the lines SA, Sa, SX, 
Sx, are proportional to SO, So : (Bk. V.Th. xii. Cor.), therefore, 

SA : Sa : : SX : Sx : 
hence, ABCDE : abede : : XYZ : xyz. 
consequently, the sections abede, xyz, are to each other as the 
bases ABCDE, XYZ. 

Cor. If the bases ABCDE, XYZ, are equivalent, any sec- 
tions abede, xyz, made at equal distances from the bases, will 
be also equivalent 



BOOK. VI 



139 



Of the Pyramid 



THEOREM VI. 

The convex surface of a right pyramid is equal to halfth pro- 
duct cf the perimeter of its base multiplied by the slant height. 

Let S— ABODE he a right pyra- 
mid, SF its slant height : then will its 
convex surface be equal to half the 
product 

8Fx(AB+BC+CD+DE+EA). 

For, since the pyramid is right, the 
point O, in which the axis meets the 
base, is the centre of the polygon 
ABODE; hence, the lines 0.4, OB, 
dec drawn to the vertices of the base, 
are equal (Bk. IV. prob. x.Cor). 

Now, in the right angled triangles SAO, SBO, the bases 
and perpendiculars are equal : hence, the hypothenuses are 
equal ; and in the same way it may be proved that all the 
edges of the pyramid are equal. The triangles, therefore,, 
which form the convex surface of the prism, are all equal tn 
ftach other. 

But the area of either of these triangles, as SAB, is equal 
to half the product of the base AB, by the slant height of the 
pyramid SF: hence, the area of all the triangles, which form 
the convex surface of the pyramid, is equal to half the product 
of the perimeter of the base by the slant height. 




THEOREM VII. 



The convex surface of the frustum of a regular pyramid is 
equal to half the sum of tl/e perimeters of the upper and lower 
bases multwlied by the slant height. 



140 



GEOMETRY. 



Of the Cone. 




Let a — ABCDE be the frustum of a 
regular pyramid : then will its convex 
surface be equal to half the product of 
the perimeter of its two bases multi- 
plied by the slant height Ff 

For, since the upper base abcde, is 
similar to the lower base ABCDE 
(Th. iv), and since ABCDE is a regular polygon, it follows 
that the sides ab, be, cd, de, and ea, are all equal to each other. 

Hence, the trapezoids EAae, ABba, &c, which form the 
convex surface of the frustum are equal. But the perpen- 
dicular distance between the parallel sides of these trapezoids 
is equal to Ef the slant height of the frustum. 

Now, the area of either of the trapezoids, as AEea, is equal 
to half the product of Ffx{EA + ea) (Bk. IV. Th. x): hence, 
the area of all of them, that is, the convex surface of the 
frustum, is equal to half the sum of the perimeters of the 
upper and lower bases, multiplied by the slant height. 



THEOREM VIII. 

The convex surface of a cone is equal to half the product of th<> 
circumference of the base multiplied by the slant height. 

In the circle which forms the base 
of the cone, inscribe a regular poly- 
gon, and join the vertices with the 
vertex S, of the cone We shall 
then have a right pyramid in- 
scribed in the cone. 

The convex surface of this pyra- 
mid will be eoual to half the product 




BOOK VI. 



14) 



Of the Cone. 



of the perimeter of the base by the 
slant height (Th. vi). 

Let us now suppose the number 
of sides of the polygon to be indefi- 
nitely increased : the polygon will 
then coincide with the base of the 
cone, the pyramid will become the 
cone, and the line Sf which meas- 
ures the slant height of the pyramid, 
will then measure the slant height 
of the cone. 

Hence, the convex surface of the cone is equal to half the 
product of the slant height by the circumference of the base. 




THEOREM IX. 

The convex surface of the frustum of a cone is equal to half 
the sum of the circumferences of its two bases multiplied by tin 
slant height. 

For, if we suppose the frustum of 
a right pyramid to be inscribed in 
the frustum of a cone, its convex 
surface will be equal to half the pro- 
duct of its slant height by the perim- 
eters of its two bases. But if we 
increase the number of sides of the 

polvgon indefinitely, the frustum of the pyramid will become 
\te frustum of the cone : hence, the area of the frustum of the 
cone is equal to half the sum of the circumferences of its twe 
beses multiplied by the slant height 




<42 



GEOMETRY 



Of Parallelopipedon 



THEOREM X. 

Two rectangular parallelopipedons, having equal altitudes and 
equal bases, are equal. 

Let E — ADCD, and F — KG HI, be two rectangular [jar 
ullelopipedons having equal % p 

bases, AC and KH, and equal 
altitudes, AE and KF : then 
will they be equal. 

For, apply the base of the 
one parallelopipedon to that B C G R 

of the other, and since the bases are equal, they will coincide 

Again, since the edges are perpendicular to the bases, the 
edges of the one parallelopipedon will coincide with those of 
the other; and since the altitude AE is equal to KF, the 
planes of the upper bases will coincide. Hence, the paral- 
lelopipedons will coincide, and consequently they are equal 



D 



KS 



\ 


\ 




T 

\ 



THEOREM XI. 

Two rectangular parallelopipedons, which have the same base, are 
to each other as their altitudes. 

Let the parallelopipedons AG, AL, 
have the same base BD, then will they 
be to each other as their altitudes AE 
A I. 

Suppose the altitudes AE, A I, to 
be to each other as two whole num- 
bers, as 15 is to 8, for example. Di- 
vide AE into 15 equal parts, whereof 
Al will contain 8; and through a?, y, ,s, 
&c, the points of division, draw planes 







BOOK VI 



143 



Of Parallelo pipe dons. 



parallel to the base. These planes 
will cut the solid A G into 15 partial 
parailelopipedons, all equal to each 
Otlicr, because they have equal bases 
and equal altitudes — equal bases, since 
every section, IL, made parallel to 
the base BD, of a prism, is equal 
to that base ; equal altitudes, because 
the altitudes are the equal divisions Ax, 
xy,yz, &c. But of these 15 equal par- 
ailelopipedons, 8 are contained in AL; 
hence, solid AG : solid AL : : 
or generally, 

solid AG : solid AL : : 



ff 

x i 



15 



AE 



& 



D 



V? 



AL 



THEOREM XII. 



Two regular parailelopipedons, having the same altitude, are to 
each oilier as their bases. 



Let the parailelopipe- 
dons AG, AK, have the 
same altitude AE ; then 
will they be to each 
other as their bases AC, 
AN. 

Having placed the two 
solids by the side of each 
other, as the figure re- 
presents, produce the 
plane ONKL until it 
meets the plane DCGH 
in PQ ; you will thus 



i 



E 



\ 



"OV 



liio 



144 



GEOMETRY 



Of Parallolopipedona 




have a third parallelo- 
pipedon A Q, which may 
be compared with each 
of the parallelopipedons 
AG,AK. The two sol- 
ids AG, AQ, having the 
same base AEHD, are 
£o each other as their 
altitudes AB, A O ; in 
like manner, the two 
solids A Q AK, having 
the same base AOLE, 
are to each other as their 
altitudes A D, AM. 

Hence, we have the two proportions, 

sohd AG : solid AQ :: AB : AO, 
solid AQ : solid AK : : AD : AM. 
Multiplying together the corresponding terms of these pro- 
portions, and omitting the common multiplier solid A Q, we have 

solid AG : solid A K :: ABxAD : AO x AM. 
But AB x AD represents the base ABCD ; and AOxAM 
represents the base AMNO: hence, two rectangular parallel- 
opipedons of the same altitude are to each other as their bases. 



THEOREM XIII. 

Any two rectangular parallelopidedons are to each other as tlu 
products of their three dimensions. 

For, having placed the two solids AG, AZ, (see next figure) 
so that their surfaces have the common angle BAE, produce 
the planes necessary for completing the third parallelopipedon 
AK, having the same altitude with the parallelopipedon AG 
By the last proposition we shall have the proportion, 






BOOK V 



145 



Of Parallelopipedons 



solid AM 



solid AK : : AB CD : AMNO 




But the two paral- 
lelopipedons AK, AZ, 
having the same base 
AMNO, are to each 
other as their altitudes 
AE, AX ; hence, we 
have 



solid AK : solid AZ : : AE : AX 

Multiplying together the corresponding terms of these pro- 
portions, and omitting in the result the common mullipliei 
solid AK, we shall have 

solid AG solid AZ : : ABCDxAE : AMNOxAX. 

Instead of the bases A BCD and AMNO, put ABxAP 
and A O x AM, and we have 

solid A : solid AZ • : ABxADxAE : A X AM X A X. 

Hence, any two rectangular parallelopipedons are to each 
other as the product of their three dimensions. 

Si-k. We are consequently authorized to assume, as the 
measure of a rectangular parallelopipedon, the product, of its 
three dimensions. 

In order to comprehend the nature of this measurement, i: 

i? neccr-sarv to reflect, that the number o( linear units in one 
13 



146 



GEOMETRY. 



Of Parallelopipedons 



dimention of the base multiplied by the number of linear unit? 
of the other dimension of the base, will give the number of 
superficial units in the base of the parallelopipedon (Bk. IV 
Th. vi. Sch). For each unit in height, there are evidently jw 
many solid units as there are superficial units in tho base. 
Therefore, the product of the number of superficial units in the 
base multiplied by the number of linear units in the altitude 
is the number of solid units in the parallelopipedon. 

If the three dimensions of another parallelopipedon are valued 
according to the same linear unit, and multiplied together in 
tho same manner, the two products will be to each other as 
the solids, and will serve to express their relative magnitude 

Let us illustrate this by an example. 

Let ABCD be the base of a 
parallelopipedon, and suppose 
AB = 4 feet, and BC = 3 feet. 
Then the number of square feet 
m the base ABCD will be equal 
to 3x4 = 12 square feet 

Therefore, 12 equal cubes of 1 
foot each, may be placed by the 
side of each other on the base. If the parallelopipedon be J 
foot in height, it will contain 12 cubic feet ; were it 2 feet in 
height, it would contain two tiers of cubes, or 24 cubic feet ; 
were it 3 feet in height, it would contain three tiers of tubes, 
or 36 cubic feet. 

The magnitude of a solid, its volume or extent, forms what 
is called its solidity ; and this word is exclusively employed 
to designate the measure of a solid ; thus, we say the s )Iidity 
of a rectangular parallelopipedon is equal to the product of it? 
base by its altitude, or to the product of its three dimensions 







BOOK VI. 



14' 



Of Parali e 1 op ip edons . 



As the cube has all its three dimensions equal, il the side 
is 1, the solidity will be 1x1x1 = 1; if the side is 2, the 
solidity will be 2x2x2 = 8; if the side is 3, the solidity 
will be 3x3 X 3 = 27; and so on: hence, if the sides of a 
series of cubes are to each other as the numbers 1, 2, 3 Sic. 
the cubes themselves, or their solidities, will be as the num- 
bers 1, 8, 27, &c. Hence it is, that in arithmetic, the cube of 
a number is the name given to a product which results from 
three factors, each equal to this number. 

THEOREM XIV. 

If a parallelopipedon, a prism, and a cylinder, have equivalent 
bases and equal altitudes, they will be equivalent. 

Let F—ABCD, be a parallelopipedon ; F— ABODE, a 
prism ; and D — ABC, a cylinder, having equivalent bases 
and equal altitudes : then will they be equivalent. 




D 




B C 



For, since their bases are equivalent they will contain the 
same number of units of surface (Bk. IV. Def. 9). Now, 
for each unit of height there will be one tier of equal cubes 
in each solid, and since the altitudes are equal, the number of 
tiers in each solid will be equal : hence, the solidities will be 
equal, and therefore the solids will be equivalent. 

Cjr Hence, we conclude, that the solidity of a prism or 
cylinder is equal tj the area of its base multiplied by its 
altitude. 



MS 



GEOMETRY. 



Of Triangular Pyramids 



THEOREM XV. 



Two triangular 'pyramids, having equivalent bums and equal 
altitudes, are equivalent, or equal in solidity. 





Let their equivalent bases, ABC, abc, be situated in the 
same plane, and let AT be their common altitude. If they 
are not equivalent, let S — abc be the smaller ; and suppose 
Aa to be the altitude of a prism, which, having ABC for its 
base, is equal to their difference. 

Divide the altitude AT into equal parts Ax, xy, yz, &c. 
each less than Aa, and let k be one of those parts : through 
the points of division pass planes parallel to the plane of the 
bases : the corresponding sections formed by these planes in 
the two pyramids will be respectively equivalent, namely 
DEF to def, GUI to ghi, &c (Th. v. Cor.) 



BOOK VI. 149 



Of Triangular Pyra m ids. 



This being granted, upon the triangles ABC, DEF, G///, 
&c, taken as bases, construct exterior prisms having foi 
edges the parts AD, DG, GR, &c, of the edge SA ; in like 
manner, on bases def, ghi, klm, &c , in the second pyramid 
construct interior prisms, having for edges the corresponding 
parts of Sa. It is plain that the sum of the exterior prisms o\ 
the pyramid S — ABC will be greater than the pyramid; while 
the sum of the interior prisms of the pyramid *S — abc, will be 
less than the pyramid. Hence, the difference between these 
sums will be greater than the difference between the pyramids. 

Now, beginning with the bases ABC, abc, the second ex- 
terior prism EFD — G is equivalent to the first interior prism 
efd — a, because they have the same altitude k, and their bases 
DEF, def, are equivalent ; for like reasons, the third exterior 
prism HIG — K, and the second interior prism hig — d, are 
equivalent ; the fourth exterior and the third interior ; and so 
on, to the last of each series. Hence, all the exterior prisms 
of the pyramid *S — ABC, excepting the first prism BCA — D, 
have equivalent corresponding ones in the interior prisms 01 
the pyramid 5 — abc: hence, the prism BCA — D is the differ- 
ence between the sum of all the exterior prisms of the pyramid 
S — ABC, and of the interior prisms of the pyramid S — abc 
But this difference has already been proved to be greater than 
that of the two pyramids : which, by supposition, differ by 
the prism a — ABC: hence, the prism BCA — D, must be 
greater than the prism a — ABC. But in reality it is less, foi 
they have the same base ABC, and the altitude Ax, of the 
first, is less than Aa, the altitude of the second. Hence, the 
supposed inequality between the two pyramids cannot exiat; 
hence, the two pyramids; <S — ABC, S — abc, having equal ai 
titudes and equivalent bases, are themselves equivalent 
18* 



150 



GEOMETRY. 



Of Triangular Pyramid 




THEOREM XVI. 

Every triangular pyramid is a third part of a tmangulai prism 
which has an equal base and the same altitude. 

Let F — ABC be a trian- 
gular pyramid, ABC — DEF 
a triangular prism of the 
same base and the same al- 
titude : the pyramid will be 
equal to a third of the prism. 

Cut off the pyramid F — 
ABC from the prism, by the 
plane FAC ; there will re- 
main the solid F — ACDE, 
which may be considered 
as a quadrangular pyramid, whose vertex is F, and whose 
base is the parallelogram ACDE. Draw the diagonal CE, 
and pass the plane FCE, which will cut the quadrangular 
pyramid into two triangular ones, F-ACE, F-CDE. These 
two triangular pyramids have for their common altitude the 
perpendicular let fall from F on the plane ACDE; and 
their bases are also equal, being halves of the parallelogram 
AD: hence, the pyramid F-ACE, and the pyramid F-CDE, 
are equivalent (Th. xv). 

But the pyramid F — CDE, and the pyramid F — ABC, have 
equal bases, ABC, DEF ; they have also the same altitude, 
namely, the distance between the parallel planes ABC, DEF, 
hence, the two pyramids are equivalent. Now, the pyramid 
F — CDE has already been proved equivalent to F — ACE ; 
hence, the three pyramids F— ABC, F—CDE, F—ACE, 
which compose the prism ABC — DEF are all equivalent 



BOOK VI 



151 



Soliility of the Pyramid. 

Hence, the pyramid F — ABC is the third part of the prism 
ABC — DEF, which has an equal base and the same altitude. 

Car. The solidity of a triangular pyramid is equal to a third 
[•art of the product of its base by its altitude. 

THEOREM XVII. 

The solidity of every pyramid is equal tc the base multiplied fry 
a third of the altitude. 

Let S — ABCDE be a pyramid. 

Pass the planes SEB, SEC through 
the diagonals EB, EC ; the polygonal 
pyramid S— ABCDE will be divided 
into several triangular pyramids all 
having the same altitude SO. But 
each of these pyramids is measured by 
multiplying its base ABE, BCE, or 
CDE, by the third part of its altitude 
SO (Th. xvi. Cor.); hence the sam 
of these triangular pyramids, or the polygonal pyramid 
S — ABCDE, will be measured by the sum of the triangles 
ABE, BCE, CDE, or the polygon ABCDE, multiplied by 
one third of SO. 

Cor. 1. Every pyramid is the third part of the prism which 
is the same base and the same altitude. 

Cor. 2. Two pyramids having the same altitude, are to 
each other as their bases. 

Cor. 3. Two pyramids having equivalent bases, are to each 
other as their altitudes. 

Co?. 4. Pyramids are to each other as the products of their 
bases by their altitudea 




52 GEOMETRY 



Sjlidity of the Cone 




THEOREM XVIII. 

The solidity of a cone cs equal to one third of the product of the 
base multiplied by the altitude. 

Let ABCDE be the base, S the 
vertex, and SO the altitude of the 
cone : then will its solidity be equal 
to one third the product of its base 
by its altitude SO. 

Inscribe in the base of the cone 
any regular polygon, ABCDE, and 
join the vertices A, /i, C, &c, with 
the vertex S, of the cone ; then will B 

there be inscribed in the cone a right pyramid, having 
for its base the polygon ABCDE. The solidity of this 
pyramid is equal to one third of the base multiplied by the 
altitude (Th. xvii). 

Let now, the number of sides of the polygon be indefinitely 
increased : the polygon will then become equal to the circle, 
and the pyramid and cone will coincide and become equal. 
But the solidity of the pyramid will still be equal to one third 
of the product of the base multiplied by the altitude, whatever 
be the number of sides of the polygon which forms its base ; 
hence, the solidity of the cone is equal to one third of the 
product of its base multiplied by its altitude. 

Cor. 1. A cone is the third part of a cylinder having the 
same base and the same altitude ; whence it follows : 

1st, That cones of equal altitudes are to each other as theti 
bases. 

2nd, That cones of equal bases are to each other a«? their 
altitudes. 



BOOK VI 



153 



Of PrisniP. 



Cor. 2. The solidity of a cone is equivalent to the solidity 
of a pyramid having an equivalent base and the same altitude. 

THEOREM xix. 

Similar prisms are to each other as the cubes of their homologous 

edges. 



Let ABC—D, EFG—H be 
similar prisms : then we shall 
havo 



./ 




B_C 
solid AD : solid EH : : AB 3 

or solid AD : solid EH : : CD 

or, the solids will be to each other as the cubes of any othei 
of their homologous edges. 

For, the solids are to each other as the products of theii 
bases and altitudes (Th. xiv. Cor.), that is, 

sdid ABC-D : solid EFG-H : : ABC x CD : EFGxGH. 

But the bases being similar polygons are to each other as the 
squares of their like sides (Bk. IV. Th. xxi) ; that is, 



ABC : EFG : : 
therefore, 
solid ABO-D : solid EFG-H 



AB 



EF\ 
AffxCD : EfxGH. 



54 



GEOMETRY. 



Of Prisms 



But since the solids are simi- 
lar, the parallelograms BD and 
FH are similar (Def. 3) : hence, 
CD and GH are proportional to 
BC and FG, and consequently 
to .47? and EF: hence, we have, 



<D 



/ 



V 



\ 



irfc f 



G 



: Atf'x.AZ? : EF-xEF. 



EF 6 ; 



solid ABC-D : solid EFG-H 
that is, 

wfcj ABC-D : so/id EFG-H : : H 3 
and in a similar manner it may be shown that the solids 
are to each other as the cubes of any other homologous edges. 

Cor. Since cylinders are to each other as the product of 
their bases and altitudes (Th. xiv. Cor.), it follows that similar 
cylinders are to each other as the cubes of the linear dimen- 
sions. 

THEOREM XX. 

Every section of a sphere, made by a plane, is a circle. 

Let AMB be a section, made by 
a plane, in the sphere whose cen- 
tre is C. 

From the centre C draw CO, 
perpendicular to the plane AMB, 
and also draw the lines CA, CM, 
&c, to the points of the curve 
AMB, which terminate the sec- 
tion, and join OA s OM, &c. 




BOOK 



155 



Of the Sphere 




Then, since CO is perdendic- 
ular to the plane AMB, the an- 
gles COA, COM &c, will be 
right angles, and since the radii 
of the sphere are all equal, the 
right angled triangles CAO, COM, 
&c, will have the hypothenuses 
equal, and the side CO common : 
hence, the remaining sides will be equal (Bk. I. Th. xixj. 
Therefore, all lines drawn from O to any point of the curve 
AMB are equal : hence AMB is a circle. 

Cor. 1. If the section passes through the centre of the 
sphere, its radius will be the radius of the sphere : hence, all 
great circles are equal. 

Cor. 2. Two great circles always bisect each other ; for 
their common intersection, passing through the centre, is a 
diameter. 

Cor. 3. Every great circle divides the sphere and its sur- 
face into two equal parts : for, if the two hemispheres were 
separated id afterwards placed on the common base, with 
their convexities turned the same way, the two surfaces would 
exactly coincide, no point of the one being nearer the centre 
than any point of the other. 

Cor. 4. The centre of a small circle, and that of the sphere, 
are in the same straight line, perpendicular to the plane of the 
small circle 



Cor 5. Small circles are the less the farther they lie from 



156 



GEOM E T K V 



Of the Spher 



the centre of the sphere ; for the greater CO is, the less is 
the chord AB, the diameter of the small circle AMB. 



THEOREM XXI. 

Every plane perpendicular to a radius rj its extremity is tan- 
gent to the sphere. 

Let FAG be a plane perpen- 
dicular to the radius OA, at its 
extremity A. Any point M, in 
this plane, being assumed, and 
OM, AM, being drawn, the an- 
gle OAM will be a right angle, 
and hence, the distance OM will 
be greater than OA. Hence, 
the point M lies without the sphere ; and as the same can be 
shown for every other point of the plane FAG, this plane can 
have no point but A common to it and the surface of the 
sphere ; hence it is a tangent plane (Def. 31). 

Sch. In the same way it may be shown, that two spheres 
have but one point in common, and therefore touch each 
other, when the distance between their centres is equal to the 
sum, or the difference of their radii ; in either case, the 
centres and the point of contact lie in the same straight line. 




THEOREM XXII. 

If a regular semi-polygon he revolved about a line passing 
though the centre and the vertices of two opposite angles, the 
surface described by its perimeter will be equal to tlie axis -multi 
plied by Uie circumference of the inscribed cvrclr.. 



BOOK VI. 



157 



Of ihe Sphere 




Suppose the regular semi-polygon 
ABCDE to be revolved about the line 
AF as an axis: then will the surface 
described by its perimeter be equal to 
AF multiplied by the circumference of 
the inscribed circle. 

From E and D, the extremities of 
one of the equal sides, let fall the per- 
pendiculars EH, DI, on the axis AF, 
and from the centre O, draw ON per- 
pendicular to the side BE: ON will then be the radius of the 
inscribed circle (Bk. IV. Prob. x). 

Let us first find the measure of the surface described by 
one of the equal sides, as DE. 

From N, the middle point of DE, draw NM perpendicular 
to the axis AF, and through E, draw EK, parallel to it, meet- 
ing MN in 5. 

Then, since EN is half of ED, NS will be half of DK 
(Bk. IV. Th. xiii) : and hence, NM is equal to half the sum 
of EH+DI. 

But, since the circumferences of circles are to each other as 
their diameters (Bk. IV. Th. xxiv), or as their radii, the 
halves of the diameters, we shall have the circumference de- 
scribed by the point A 7 , equal to half the sum of the circum- 
ferences described by the points D and E. 

But in the revolution of the polvgon the line ED describes 
the surface of the frustum of a cone, the measure of which is 
equal to DE multiplied into half the sum of the circumfe- 
rences of the two bases (Th. ix) ; that is, equal to DE into 

the circumference described bv the point N 
14 



158 



GEOMETRY. 



Of the S p h e r e 




But, the triangle ENS is similar to 
SNT (Bk. IV. Th. xviii), and also to 
EDK, and since TNS is similar to 
ONM, it follows that EDK and ONM 
are similar ; hence, 



ED : EK or HI : : ON : NM, 
or ED : HI : : circumference ON : circumference MN. 
consequently, 

ED X circumference MN= HI X circumference ON, 

that is, ED multiplied into the circumference of the circle de- 
scribed with tbe radius NM, is equal to HI into the circum- 
ference of the circle described with the radius ON. But the 
former is equal to the surface described by the line ED in the 
revolution of the polygon about the axis AF; hence, the latter 
is equal to the same area ; and since the same may be shown 
for each of the other sides, it is plain that the surface des- 
cribed by the entire perimeter is equal to 
{FH-\-HI+IP+PQ+QA)xcii>f. ON=AFxcirf. ON. 

Cor. The surface described by any portion of the perim- 
eter, as EDO, is equal to the distance between the two per- 
pendiculars let fall from its extremities, on the axis, multiplied 
by the circumference of the inscribed circle. For, the sur- 
face described by DE is equal to HIx circumference ON 
and the surface described by DC is equal to /P X circumfe- 



BUOK VI 



159 



Of the Sphere 



nmce ON: hence, the surface described by ED+DC, is equal 
co {111+ IP)X circumference ON, or equal to HP xcu cum- 
Terence ON. 




THEOREM XXIII. 

The surface of a sphere is equal to the product of its diameter 
by the circumference of a great circle. 

Let ABODE be a semicircle. In- 
scribe in it any regular semi-polygon, 
and from the centre O draw OF per- 
pendicular to one of the sides. 

Let the semicircle and the semi- 
polygon be revolved about the axis 
AE: the semicircumference ABODE 
will describe the surface of a sphere 
(Def. 26) ; and the perimeter of the 
semi-polygon will describe a surface 
which has for its measure AEx cir- 
cumference OF (Th. xxii) ; and this will be true whatever be 
the number of sides of the polygon. But if the number of 
sides of the polygon be indefinitely increased, its perimeter 
will coincide with the circumference ABODE, the perpen- 
dicular OF will become equal to OE, and the surface do- 
Bcribed by the perimeter of the semi-polygon will then be the 
same as that described by the semicircumference ABODE 
Hence, the surface of the sphere is equal to AE x circum 
ference OE. 

Cot. Since the area of a great circle is equal to the product 
oi its circumference by half the radius, or by one-fourth of 
the diameter (Bk. IV. Th. xxvii), it follows that the surface 
of a sphere is equal to four of its great circles. 



160 



3 E O M E T R Y 



Of the Zone 



THEOREM XXIV. 

The surface of a zone is equal to its altitude multiplied by 
the circumference of a great circle. 

For, the surface described by any 
portion of the perimeter of the in- 
scribed polygon, as BC-\-CD is equal 
to EH x circumference OF (Th. xxii. 
Cor). But when the number of sides 
of the polygon is indefinitely increased, 
BC+CD, becomes the arc BCD, OF 
becomes equal to OA, and the surface 
described by BC+CD, becomes the 
surface of the zone described by the 
arc BCD: hence, the surface of the 
zone is equal to EHx circumference 
OA. 

Sch. 1. When the zone has but one base, as the zone dc 
scribed by the arc ABCD, its surface will still be equal to 
the altitude AE multiplied by the circumference of a greal 
circle. 

Sch. 2. Two zones taken in the same sphere, or in equal 
spheres, are to each other as their altitudes ; and any zone is 
to the surface of the sphere as the altitude of the zone is to 
the diameter of the sphere. 




THEOREM XXV. 



The solidity of a sphere is equal to one third of the product if 
the surface multiplied by the radius. 
For, conceive a polyedron to be inscribed in the sphere. 



BOOK VI. 



L6J 



Of the Sphere 



This polyedron may be considered as formed of pyramids, each 
having for its vertex the centre of the sphere, and for its base 
one of the faces of the polyedron. Now, the solidity of each 
pyramid, will be equal to one third of the product of its base 
i>y its altitude (Th. xvii). 

But if we suppose the faces of the polyedron to be continu- 
ally diminished, and consequently, the number of the pyra- 
mids to be constantly increased, the polyedron will finally 
become tlie sphere, and the bases of all the pyramids will 
become the surface of the sphere. When this takes place, 
the solidities of the pyramids will still be equal to one third 
tha product of the bases by the common altitude, which will 
then be equal to the radius oi^ the sphere. 

Hence, the solidity of a sphere is equal to one third of the 
product of the surface bv the radius. 



THEOREM XXVI. 

The surface of a sphere is equal to the convex surface of the 
circumscribing cylinder ; and the solidity of the sphere is two 
thirds the solidity of the circumscribing cylinder. 

Let MPNQ be a great circle of 
the sphere ; ABCD the circum- 
scribing square : if the semicircle 
PMQ, and the half square PADQ, 
are at the same time made to re- 
volve about the diameter PQ, the 
semicircle will describe the sphere, 
vhile the half square will describe 
the cylinder circumscribed about 
that sphere. 

The altitude AD, of the cylinder, is equal to the diameter 
14* 




162 



GEOMETRY 



Of the Sphere. 







PQ; the base of the cylinder is 
equal to the great circle, since its 
diameter A B is equal MN; hence, 
the convex surface of the cylin- 
der is equal to the circumference 
of the great circle multiplied by 
its diameter (Th. ii). This meas- 
ure is the same as that of the sur- 
face of the sphere (Th. xxiii) : 
hence the surface of the sphere is equal to the convex sur- 
face of the circumscribing cylinder. 

In the next place, since the base of the circumscribing 
cylinder is equal to a great circle, and its altitude to the di- 
ameter, the solidity of the cylinder will be equal to a great 
circle multiplied by a diameter (Th. xiv. Cor). But the so- 
lidity of the sphere is equal to its surface multiplied by a third 
of its radius ; and since the surface is equal to four great 
circles (Th. xxiii. Cor.), the solidity is equal to four great cir- 
cles multiplied by a third of the radius ; in other words, to 
one great circle multiplied by four-thirds of the radius, 01 
by two-thirds of the diameter, hence, the sphere is two-thirds 
of the circumscribing cylinder. 



BOOK VI 



163 



Appendix 



APPENDIX 
OF THE FIVE REGULAR PCLYEDRONS. 

A regular polyedron, is one whose faces are all equal poly- 
gons, and whose polyedral angles are equal. There are fiVe 
such solids. 

1. The Tetraedron, or equilateral pyramid, is a solid bounded 
by four equal triangles. 




2. The hexacdron or cube, is a solid, bourded by six equal 
squares. 




3. The oclaedrrm, is a soiid, bounded by eight equal e.ftu 



lateral triangles. 




104 



GEOMETRY. 



Appendix 



4. The dodeecudron, is a solid bounded by twelve oquaJ 

pentagons 




5. The icosacdron, is a solid, bounded by twenty equa 
equilateral triangles. 




G. The regular solids may easily be made of pasteboard. 

Draw the figures of the regular solids accurately on paste 
board, and then cut through the bounding lines : this will give 
figures of pasteboard similar to the diagrams. Then, cut 
the other lines half through the pasteboard, after which, turn 
up the parts, and glue them together, and you will form the 
bodies which have been described. 



ELEMENTS OF TRIGONOMETRY. 



INTRODUCTION. 
SECTION I. 

OF LOGAniTOMS. 



1. The logarithm of a number is the exponent of the \iowcr 
to which it is necessary to raise a fixed number, in order to 
produce the first number. 

This fixed number is called the base of the system, and maj 
beany number except 1 : in the common system 10 is assumed 
as the base. 

2. Tf we form those powers of 10, which are denoted by entire 
exponents, we shall have 

10° = 1 10 1 = 10 , 10 3 =r 1000 

10 2 = 100 , 10 4 = 10000, &c. &c. 
From the above table, it is plain, that 0, 1, 2, 3, 4, <fec, are re- 
spectively the logarithms of 1, 10, 100, 1000, 10000, &c. ; we 
also see that tho logarithm of any number between 1 and 10 is 
greater than -nd less than 1 : thus 
Log 2^0.201030 



166 TRIGONOMETRY. 



Of Logarithms, 



The logarithm of any number greater than 10, and less than 
100, is greater than 1 and less than 2 : thus 
Log 50 = 1.698970 

The logarithm of any number greater than 100, and less than 
1000, is greater than 2 and less than 3 : thus 
Log 126 — 2.100371, &c. , 

Tf the above principles be extended to other numbers, it will 
appear, that the logarithm of any number, not an exact power 
often, is made up of two parts, an entire and a decimal "part 
The entire fart is called the characteristic of the logarithm, 
and is always one less than the number of places of figures in the 
given number. 

3. The principal use of logarithms, is to abridge numerical 
computations. 

Let M denote any number, and let its logarithm be denoted 
by m ; also let N denote a second number whose logarithm is 
n / then from the definition we shall have 

10 m = M (1) 10° = N (2) 

Multiplying equations (1) and (2), member by member, we 
have 

10 m+ * = MxN or, m+n = log MxN : hence, 

The sum of the logarithms of any two numbers is equal to 
the logarithm of their product. 

Dividing equation (1) by equation (2), member by member, 
we have 

, m-n M . M . 

10 = — - or, m — n = Io£ rr=.: hence, 

The logarithm of the quotient of two numbers, is equal to tht 
logarithm of the dividend diminished by the logarithm of the 
divisor. 



INTRODUCTION. 167 



Of Logarithms, 



4. Since the logarithm of 10 is 1, the logarithm of the product 
of any number by 10, will be greater by 1 than the logarithm oj 
that number ; also, the logarithm of any number divided by \ f 
will be less by 1 than the logarithm of that number. 

Similarly, it may be shown that the logarithm of any number 
multiplied by a hundred, is greater by 2 than the logarithm of 
that number, and the logarithm of any number divided by 100 
b less by 2, than the logarithm of that number, and so on. 

EXAMPLES. 

log 327 is 2.514548 

log 32.7 " 1.514548 

log 3.27 " 0.514548 

log .327 " 1.514548 

log .0327 " 2.514548 

From the above examples, we see, that in a number composed 
of an entire and decimal part, we may change the place of the 
decimal point without changing the decimal part of the logarithm, 
but the characteristic is diminished by 1 for every place that the 
decimal point is removed to the left. 

In the logarithm of a decimal, the characteristic becomes nega- 
tive, and is numerically 1 greater than the number of ciphers im- 
mediately after the decimal point. The negative sign extends 
only to the characteristic, and is written over it as in the exam- 
ples given above. 

TABLE OF LOGARITHMS. 

5. A table of logarithms, is a table in which are written the 
logarithms of all numbers between 1 and some given number, 
The logarithms of all numbers between 1 and 10,000 are giveD 



IOH TRIGONOMETRY. 



Of Logarithms. 



in the annexed table. Since rules have been given for determin- 
ing the characteristics of logarithms by simple inspection, it has 
not been deemed necessary to write them in the table, the deci- 
mal part only being given. The characteristic, however, is given 
for all numbers less than 100. 

The left hand column of each page of the table, is the column 
of numbers, and is designated by the letter N ; the logarithms 
of these numbers are placed opposite them on the same hori- 
zontal line. The last column on each page, headed I), shows the 
difference between the logarithms of two consecutive numbers. 
This difference is found by subtracting the logarithm under the 
column headed 4, from the one in the column headed 5 in the 
aame horizontal line, and is nearly a mean of the differences 
of any two consecutive logarithms on the line. 

6. To find from the table the logarithm of any number. 

If the number is less than 100, look on the first page of the 
cable, in the column of numbers under N, until the number is 
found : the number opposite is the logarithm sought: Thus 
log 9 = 0.954243 

7. When the number is greater than 100 and less than 10000. 
Find in the column of numbers, the first three figures of the 

given number. Then pass across the page along a horizontal 
line until you come into the column under the fourth figure of 
the given number: at this place, there are four figures of the 
required logarithm, to which two figures taken from the column 
marked 0, are to be prefixed. 

If the four figures already found stand opposite a row of six 
figures in the column marked 0, the two left hand figures of 
the six, are the two to be prefixed ; but if they stand opposite 



NTRODUCiION. Kit) 



Of Logarithms. 



a row of only four figures, you ascend the column tiil ^ou find 
a row of six figures; the two left band figures of this row are 
the two to be prefixed. If you prefix to the decimal part thus 
found, the characteristic, you will have the logarithm sought: 
Thus, 

log 8979 = 3.953228 

log .08979 = 2.953228 
If however in passing back from the four figures found, to the 
column, any dots be met with, the two figures to be prefixed 
must be taken from the horizontal line directly below : Thus, 

log 3098 = 3.491081 

log 30.98 = 1.491081 
If the logarithm falls at a place where the dots occur, must 
be written for each dot, and the two figures to be prefixed are 
as before taken from the line below : Thus, 

log 2188 = 3.340047 

log .2188 = 1.340047 

8. When the number exceeds 10,000. 

The characteristic is determined by the rules already given. 
fo find the decimal part of the logarithm. Place a decimal 
point after the fourth figure from the left hand, converting the 
given number into a whole number and decimal. Find the loga- 
rithm of the entire part by the rule just given, then take from 
the right hand column of the page, under D, the number on the 
same horizontal line with the logarithm, and multiply it by the 
decimal part; add the product thus obtained to the logarithm al- 
ready found, and the sum will be the logarithm sought. 

If, in multiplying the number taken from the column D, the 

decimal part of the product exceeds .5 let 1 be added to the er»- 
15 



170 TRIGONOMETRY 

Of Logarithms. 

tire part; if it is less than .5 the decimal part of the product b 
neglected. 

EXAMPLE. 

To find log 672887. 

The characteristic is 5. ; placing a decimal point after the 
fourth figure from the left, we have 6728.87. The decimal part 
of the log 6728 is .827886 and the corresponding numher in the 
column D is 65 ; then 65X-87 = 56.55, and since the decimal 
part exceeds .5, we have 57 to be added to 827886, which givea 
.827943 

or log 672887 = 5.827943 
Similarly log .0672887 = 2.827943 

The Inst rule has been deduced under the supposition that the 
difference of the numbers is proportional to the difference of 
their logarithms, which is sufficiently exact within the narrow 
limits considered. 

In the above example, 65 is the difference between the loga- 
rithm of 672900 and the logarithm of 672800, that is, it is the 
difference between the logarithms of two numbers which differ by 
100. 

We have then the proportion 100 : 87 : 65 . 56.55, the 
number to be added to the logarithm already found. 

9. To find from the table the number corresponding to a 
given logarithm. 

Search in the columns of logarithms for the decimal part of 
the given logarithm : if it cannot be found in the table, take 
out the number corresponding to the next less logarithm and 
set it aside. Subtract this less logarithm from the given loga- 
rithm, and annex to the remainder as many zeros as may be 



INTRODUCTION. 171 

Of Logarithms. 

necessary, and divide this result by the corresponding number 
taken from the column marked O, continuing the division aa 
long as desirable : annex the quotient to the number set aside. 
Point off, from the left hand, as many integer figuies as there are 
units in the characteristic of the given logarithm increased by 
1 ; the result is the required number. 

If the characteristic is negative, the number will be entirely 
decimal, and the number of zeros to be placed immediately after 
the decimal point will be equal to the number of units in the 
characteristic diminished by 1. 

This rule, like its converse, is founded on the supposition that 
the difference of the logarithms is proportional to the difference 
of their numbers within narrow limits. 

EXAMPLE. 

Find the number corresponding to the logarithm 3.233568. 

The decimal part of the given logarithm is .233568 

The next less logarithm of the table is .233504 and its 

corresponding number 1712. 

Their difference is 64 



Tabular difference 253)6400000(25 

Hence the number sought 1712.25 

The number corresponding to 3.2335G8 is .00171225 

MULTIPLICATION BY LOGARITHMS. 

10. When it is required to multiply numbers by means ot 
their logarithms, we tirst find from the table the logarithms of 
the numbers to be multiplied ; we next add these logarithms 
together, and their sum is the logarithm of the product of the 
numbers (Art. 3). 

The term sum is to be understood in its a'gebraic sense: 



172 TRIGONOMETRY. 



Of Logarithms. 



therefore, if any of the logarithms have negative characteristics, 
the difference between their sum and that of the positive 
characteristics, is to be taken ; the sign of the remainder ia 
that of the greater sum. 

EXAMPLES. 

1. Multiply 23.14 by 5.062. 

log 23.14 = 1.384363 
log 5.062 = 0.704322 



Product 117.1347 .... 2.06868; 



2. Multiply 3.902, 597.16 and 0.0314728 together, 
log 3.902 == 0.591287 
log 597.16 = 2.776091 
loo; 0.0314728 = 2.497936 



Product 73.3354 .... 1.865314 



Here the 2 cancels the -f- 2, and the 1 carried from the deoi- 
mal part is set down. 

3. Multiply 3.586, 2.1046, 0.8372, and 0.0294, together, 
log 3.586 = 0.554610 
log 2.1046 = 0.323170 
log 0.8372 = 1.922829 
loo; 0.0294 = 2.468347 



Product 0.1857615 . . 1.268956 



In this example the 2, carried from the decimal part, cancels 
2, and there remains 1 to be set down. 

DIVISION OF NUMBERS BY LOGARITHMS. 

11. When it is required to divide numbers by means of their 
logarithms, we have only to recollect, that the subtraction of 



INTRODUCTION. 173 

Of Logarithms. 

logarithms corresponds to the division of their numbers (Art. 3). 
Hence, if we find the logarithm of the dividend, and from it oub- 
tract the logarithm of the divisor, the remainder will be the loga- 
rithm of the quotient. 

This additional caution may be added. The difference of tho 
logarithms, as here used, means the alyebraic difference; so 
that, if the logarithm of the divisor have a negative characteristic 
its sign must be changed to positive, after diminishing it by the 
unit, if any, carried in the subtraction from the decimal part of 
the logarithm. Or, if the characteristic of the logarithm of the 
dividend is negative, it must be treated as a negative number. 

EXAMPLES. 

1. To divide 24163 by 4567. 

log 24163 = 4.383151 
losr 4567 = 3.059631 



Quotient 5.29078 .... 0.723520 

2. To divide 0.06314 by .007241 

log 0.0C314 = 2.800305 

log 0.007241 = 3.859799 

Quotient . . 8.7198 . . . .. 0.940506 

Here, 1 carried from the decimal part to the 3 changes it. to 
2, which being taken from 2, leaves for the characteristic. 

3 To divide 37.149 by 523.76 

log 37.149 = 1.569947 
log 523.76 = 2.719133 



Quotient . . 0.0709274 . 2.850814 
15* 



174 TRIGONOMETRY. 



Of Logaritl; 



4. To divide 0.7438 by 12.9476 

log 0.7438 == 1.871456 
log 12.9476 = 1.112189 



Quotient . . 0.057447 . . 2.759267 

Here, the 1 taken from 1, gives 2 for a result, as set down. 



ARITHMETICAL COMPLEMENT. 

12. The Arithmetical complement of a logarithm is the num 
her which remains after subtracting the logarithm from 10. 

Thus, . . 1-9.274687 = 0.725313 

Hence, 0.725313 is the arithmetical complement 

of 9.274687. 

13. We will now show that, the difference between two loga- 
rithms is truly found, by adding to the first logarithm the 
arithmetical complement of the logarithm to be subtracted, and 
then diminishing the sum by 10. 

Let a = the first logarithm 

b = the logarithm to be subtracted 
and c = 10 — b = the arithmetical complement ot 6. 

Now the difference between the two logarithms will be ex 
pressed by a — b. 

But, from the equation c = 10 — b, we have 
c-10 = —b 
hence, if we place for— b its value, we shall have 

a — b = a-\-c — 10 
which agrees with the enunciation. 

When we wish the arithmetical complement of a logarithm, 
we may write it directly from the table, by subtracting the left 



O TRODUCTIOW. 175 

Of Logarithms. 

hand fgure from 9, then proceeding to the right, subtract each 
figure from 9 till we reach the last significant figure, which 
must be taken from 10 : this will be the same an taking the 
logarithm from 10. 

EXAMPLES. 

1. From 3.274107 take 2.104729. 

By common method. By arith. comp. 

3.274107 3.274107 

2.104729 its ar. corn p. 7.895271 

Diff. 1.169378 Sum 1.169378 after sub 



tracting 10. 

Hence, to perform division by means of the arithmetical com- 
plement we have the following 

RULE. 

To the logarithm of the dividend add the arithmetical com- 
plement of the logarithm, of the divisor : the sum after subtract' 
ing in. will be the logarithm of the quotient. 

EXAMPLES. 

1. Divide 32 7.5 by 22.07. 

log 327.5 . . . 2.515211 • 

log 22.07 ar. comp. 8.656198 

Quotient . . 14.839 .... 1.171409 

2. Divide 0.7438 by 12.9476. 

log 0.7438 T.871456 

leg 12.9476 ar. comp. 8.887811 

Quotient . . 0.057447 . . . 2.759267 



I/O rRIGONOMETRY. 

Description of Instruments. 

In this example, the sum of the characteristics is £, from 
which, taking 10, the remainder is 2. 

3. Divide -37.149 by 523.76, 

log 37.149 *..... 1.560947 
log 523.76 ar. coup. 7.280867 

Quotient . . 0.0709273 . . . 2.850814 



SECTION II. 

OF SCALES. 
SCALE OF EQUAL PARTS. 

I I t i — i — r- i i i i -r-f 



14. A scale of equal parts is formed by dividing a line oi a 
^ven length into equal portions. 

If, fSr example, the line ab of a given length, say one inch, 
be divided into any number of equal parts, as 10, the scale thus 
formed, is called a scale of ten parts to the inch. The line a&, 
which is divided, is called the unit of the scale. This unit is 
laid off several times on the left of the divided line, and its 
extremities marked, 1, 2, 3, &c. 

The unit of scales of equal parts, is, in general, either an 
inch, or an exact part of an inch. If, for example, ab the unit 



INTRODUCTION. 



177 



Description of Instruments. 

of the scale, were half an inch, the scale would be cue of IG 
parts to half an inch, or of 20 paits to the inch. 

If it were required to take from the scale a line equal to two 
inches and six-tenths, place one foot of the dividers at 2 on the 
'/eft, and extend the other to .6, which marks the sixth of the 
small divisions : the dividers will then embrace the required 
distance. 



DIAGONAL SCALE OF EQUAL PARTS. 



- 


df c 






\\ \ I / / It 


J 


03 


j 1 M / / / ■ I 1 i 


n 


•08 


i ? « / 191 1 1 1 




•07 


1 / ' 1 1 I 1 1 


1 


•06 


/ / M 1 M M 1 


?! 


05 


f / W I 


l\ 


.04 


MM/// 


1 


.03 


/ / I M J / 


| -02 


null hi 


1 


inillilii 


2 

h I 


a 


.2 .3 .3.4 ,5 .6.7 .S .9 b 



J 5. This scale is thus constructed. Take ab for the unit *f 
the scale, which may be one inch, \ | or J of an inch, in length. 
On ab describe the square abed. Divide the sides ab and dc 
each into ten equal parts. Draw of and the other nine parallels 
as in the figure. 

Produce ba to the left, and lay off the unit of the scale any 
convenient number of times, and mark the points 1, 2, 3, &c. 
Then, divide the line ad into ten equal parts, and through the 
points of division draw parallels to ab as in the figure. 

Now, the small divisions of the line ab are each one-tenth 
(.1) of ab ; they are therefore .1 of ad, or .1 of ag or gh. 

If we consider the triangle adf, we see that the base df if 



178 TRIGONOMETRY 

Dosoriptioij of Instruments. 

one-tenth of ad, the unit of the scale. Since the distance from 
a to the first horizontal line above ab, is one-tenth of the dis- 
tance ad, it follows that the distance measured on that line be- 
tween ad and af is one-tenth of df : but since one-tenth of a 
tenth is a hundredth, it follows that this distance is one-hun- 
dredth (.01) of the unit of the scale. A like distance measured 
on the second line will be two-hundredths (.02) of the unit of 
the scale ; on the third, .03 ; on the fourth, .04, &c. 

If it were required to take, in the dividers, the unit of the 
scale, and any number of tenths, place one foot of the dividers 
at 1, and extend the other to that figure between a and b which 
designates the tenths. If two or more units are required, the 
dividers must be placed on a point of division further to the left. 

When units, tenths, and hundredths, are required, place one 
foot of the dividers where the vertical line through the point 
which designates the units, intersects the line which designates 
the hundredths : then, extend the dividers to that line between 
ad and be which designates the tenths : the distance so deter- 
mined will be the one required. 

For example, to take off the distance 2.34, we place one foot 
of the dividers at I, and extend the other to e : and to take off 
the distance 2.58, we place one foot of the dividers at p and ex- 
tend the other to q. 

Remark I. If a line is so long that the whole of it cannot 
be taken from the scale, it must be divided, and the parts of it 
taken from the scale in succession. 

Remark II. If a line be given upon the paper, its length 
can be found by taking it in the dividers and applying it to 
the scale. 



INTRODUCTION 



179 



Description of Instrument 



SCALE OF CHORDS 




A O 

16. if, with any radius, as AC, we describe the quadrant CD, 
and then divide it into 90 equal parts, each part is called a 
degree. 

Through C, and each point of division, let a chord be drawn, 
and let the lengths of these chords be accurately laid off on a 
scale : such a scale is called a scale of chords. In the figure, 
the chords are drawn for every ten degrees. 

The scale of chords being once constructed, the radius of the 
circle from which the chords were obtained, is known ; for, the 
chord marked 60 is always equal to the radius of the circle. A 
scale of chords is generally laid down on the scales which belong 
to cases of mathematical instruments, and is marked cho. 

To lay off, at a given point of a line, with the scale nf chords, 
an angle equal to a given angle. 

Let AB be the line, and A the given 
point. 

Take from the scale the chord of 60 de- 
grees, and with this radius, and the point 
A as a centre, describe the arc BC. Then take from the scak 




ISO 



TRIGONOMETRY 



Description of Instruments, 



Ihe chord of the given angle aav 30 degrees, and with this line 
as a radius, and B as, a. centre, describe an arc cutting BC in C 
Through A and C draw the line AC, and BAG will be the re- 
quired angle. 



SEMICIRCULAR PROTRACTOR. 
C 




17. This instrument is used to lay down, or protract angles 
It may also be used to measure angles included between lines 
already drawn upon paper. 

Tt consists of a brass semicircle ABC divided to half degrees 
The degrees are numbered from to 180, both ways; that is, 
from A to B and from B to A. The divisions, in the figure, 
are only made to degrees. There is a small notch at the mid- 
dle of the diameter AB, which indicates the centre of tie pro- 
tractor. 

GUNTERS' SCALE. 

18. This is a scale of two feet in length, on the faces of 
which a variety of scales is marked. The face on which the 



TRIGONOMETRY- 181 

Definitions. 

divisions of inches are made, contains, however, all the scales 
necessary for laying down lines and angles. These are, the 
scale of equal parts, the diagonal scale of equal parts, and the 
scale of chords, all of which have been described. 



PLANE TRIGONOMETRY. 



DEFINITIONS AND EXPLANATION OF TABLES. 



19. In every plane triangle there are six parts : three sides 
and three angles. These parts are so related to each other, that 
when one side and any two other parts are given, the remain- 
ing parts can be obtained, either by geometrical construction or 
by trigonometrical computation. 

20. Plane Trigonometry explains the methods of computing 
the unknown parts of a plane triangle, when, a sufficient num- 
ber of the six parts is given. 

21. For the purpose of trigonometrical calculation, the cir- 
cumference of the circle is supposed to be divided into 360 
tqiaal parts, called degrees ; each degree is supposed to be di- 
vided into 60 equal parts, called minutes ; and each minute into 
60 equal parts, called seconds. 

Uegrees, minutes, and seconds, are designated respectively 
16 



1K2 



TRIGONOMETRY 



Definitions. 



by the characters ° ' ". For example, ten degrees, eighteen 
minutes, and fourteen seconds, would be written 10° 18' 14" 
If two lines be drawn through the centre of the circle, at 
right angles to each other, they will divide the circumference 
into four equal parts, of 90° each. Every right angle then, as 
EOA, is measured by an arc of 90° ; every acute angle, as 
BOA, by an arc less than 90°; and every obtuse angle, as 
FOA, by an arc greater than 90°. 

22. The complement of an arc is 
what remains after subtracting the 
arc from 90°. Thus, the arc EB is 
the complement of AB. The sum of 
an arc and its complement is equal 
to 90°. 

23. The supplement of an arc is 
what remains after subtracting the 
arc from 180°. Thus, 6^ is the sup- 
plement of the arc AEF. The sum of an arc and its sup* 
plement is equal to 180°. 

24. The sine of an arc is the perpendicular let fall from one 
extremity of the arc on the diameter which passes through 
the other extremity. Thus, BD is the sine of the arc AB. 

25. The cosine of an arc is the part of the diameter inter- 
cepted between the foot of the sine and centre. Thus, OB is 
the cosine of the arc AB. 

26. The tangent of an arc is the line which touches it at 
one extremity, and is limited by a line drawn through the 
other extremity and the centre of the circle. Thus, AC is the 
tangent of the arc AB. 




TRIGONOMETR V 



183 



Definitions. 



27. The secant of an arc is ^,he line drawn from the centre 

of the circle through one extremity of the arc, and limited bj 

the tangent passing through the other extremity. Thus, 00 
is the secant of the arc AB. 



28. The four lines, BD, OD, AC, OC, depend for their 
values on the arc AB and the radius OA ; they are thus 
designated : 



L \\± 



sin AB 


for 


BD 


cos AB 


for 


OB 


tan AB 


for 


AC 


sec AB 


for 


OC 




29. If ABE be equal to a quad- 
rant, or 90°, then EB will be the 
complement of AB. Let the lines 
ET and IB be drawn perpendicular 
to OE. Then, 

ET, the tangent of EB, is called the cotangent of AB ; 
IB, the sine of EB, is equal to the cosine of AB ; 
OT, the secant of EB, is called the cosecant of AB. 
In general, if A is any arc or angle, we have, 
cos A — sin (90° — ^1) 
cot A == tan (90° — A) 
cosec A = sec (90° — A) 



30. If we take an arc ABEF, greater than 90°, its sine 
will be FH ; OH will be its cosine; AQ its tangent, and OQ 
its secant. But FH is the sine of the arc GF, which is the 
supplement of AF, and OE is its cosine : hence, the sine of 



184 TRIGONOMETRY. 

Definitions. 

an arc is equal to the sine of its supplement ; and tlie cosim 
of an arc is equal to the cosine of its supplement* 

Furthermore, AQ is the tangent of the arc AF, and OQii 
its secant : GL is the tangent, and OL the secant of the sup- 
plemental arc GF. But since AQ is equal to GL, and OQ 
to OL, it follows that, the tangent of an arc is equal to the 
tangent of its supplement ; and the secant of an arc is equal 
to the secant of its supplement.* 

Let us suppose, that in a circle of a given radius, the 
lengths of the sine, cosine, tangent, and cotangent, have been 
calculated for every minute or second of the quadrant, and 
arranged in a table ; such a table is called a table of sines and 
tangents. If the radius of the circle is 1, the table is called a 
table of natural sines. A table of natural sines, therefore, shows 
the values of the sines, cosines, tangents and cotangents of all 
the arcs of a quadrant, divided to minutes or seconds. 

If the sines, cosines, tangents, and secants are known for arcs 
less than 90°, those for arcs which are greater can be found 
from them. For if an arc is less than 90°, its supplement 
will be greater than 90°, and the values of these lines are the 
same for an arc and its supplement. Thus, if we know the 
sine of 20°, we also know the sine of its supplement 160° ; 
for the two are equal to each other. 

TABLE OF LOGARITHMIC SINES. 

31. In this table are arranged the logarithms of the nume- 
rical values of the sines, cosines, tangents, and cotangents of all 

* These relations are between the numerical values of the trigonometrical 
lines; the algebraic signs, which they have in the different quadrants, arf 
not considered 



TRIGONOMETRY 185 



Uses of the Tables. 



the arcs of a quadrant, calculated to a radius of 10,000,000,000. 
The logarithm of this radius is 10. In the first and last hori- 
zontal lines of each page, are written the degrees whose sines, 
cosines, <fcc, are expressed on the page. The vertical columns 
on the left and right, are columns of minutes. 

CA'SE 1. 

To find, in the table, the logarithmic sine, cosine, tangent, or 
cotangent of any given arc or angle. 

32. If the angle is Ipss than 45°, look for the degrees in the 
first horizontal line of the different pages : then descend along 
the column of minutes, on the left of the page, till you reach 
the number showing the minutes : then pass along the hori- 
zontal line till you come into the column designated, sine, 
cosine, tangent, or cotangent, as the case may be : the numbei 
so indicated is the logarithm sought. Thus, on page 37, foi 
19° 55' we find, 

sine 19° 55' 9.532312 

cos 19° 55' 9.973215 

tan 19° 55' 9.559097 

cot 19° 55' 10.440903 

33. If the angle is greater than 45°, search for the degrees 
along the bottom line of the different pages : then, ascend 
&long the column of minutes on the right hand side of the 
page, till you reach the number expressing the minutes : then 
pass along the horizontal line into the column designated 
tang cot, sine, or cosine, as the case may be: the number po 
pointed out is the logarithm required. 

34. The column designated sine, at the top of the page, is 

1G* 



180 TRIGONOMETRY 



Ubsb of the Tables. 



designated by cosine at the bottom ; the one designated tang, 
by cotang, and the one designated cotang, by tan<r. 

The angle found by taking the degrees at the top of the 
page and the minutes from the first vertical column on the 
laft, is the complement of the angle found by taking the de- 
grees at the bottom of the page, and the minutes traced up in 
the right hand column to the same horizontal line. There- 
fore, sine, at the top of the page, should correspond with cosine, 
at the bottom ; cosine with sine, tang with cotang, and cotang 
with tang, as in the tables (Art. 11). 

If the angle is greater than 90°, we have only to subtract it 
from 180°, and take the sine, cosine, tangent or cotangent of 
the remainder. 

The column of the table next to the column of sines, and 
on the right of it, is designated by the letter D. This column 
is calculated in the following manner. 

Opening the table at any page, as 42, the sine of 24° is 
found to be 9,609313; that of 24° 01', 9.609597: their dif- 
ference is 284 ; this being divided by 60, the number of seconds 
in a minute, gives 4.73, which is entered in the column D. 

Now, supposing the increase of the logarithmic sine to be 
proportional to the increase of the arc, and it is nearly so for 
60", it follows, that 4.73 is the increase of the sine for 1". 
Similarly, if the arc weie 24° 20' the increase of the sine for 
1", would be 4.65. 

The same remarks are applicable in respect of the column 
D, after the column cosine, and of the column D, between 
the tangents and cotangents. The column D between the 
columns tangents and cotangents, answers *to both of these 
columns. 



TRIGONOMETRY. 237 

UsosoftheTables. 

Now, if it were required to find the logarithmic sine of an 
arc expressed in degrees, minutes, and seconds, we have only 
to find the degrees and minutes as before ; then, multiply the 
corresponding tabular difference by the seconds, and add the pro- 
duct to the number first found, for the sine of the given arc 

Thus, if we wish the sine of 40° 26' 28". 

The sine 40° 26' 9.811952 

Tabular difference 2.47 .... 

Number of seconds 28 . 



Product . . 69.16 to be added 69.16 

Gives for the sine of 40° 26' 28" 9.812021. 



The decimal figures at the right are generally omitted in 
the final result ; but when they exceed five-tenths, the figure on 
the left of the decimal point is increased by 1 ; this gives the 
nearest approximate result. 

The tangent of an arc, in which there are seconds, is found 
in a manner entirely similar. In regard to the cosine and co- 
tangent, it must be remembered, that they increase while the 
arcs decrease, and decrease as the arcs are increased ; conse- 
quently, the proportional numbers found for the seconds, must 
be subtracted, not added. 

EXAMPLES. 

1. To find the cosine of 3° 40' 40" 

The cosine of 3° 40' ... 9.99911.0 

Tabular difference .13 

Number of seconds 40 . 

Product 5.20 to be subtracted 5.20 

Gives for the cosine of 3° 40' 40" . 9.999105 



188 TRIGONOMETRY 



D sea of the Tablos. 



2. Find the tangent of 3*7° 28' 31" 

3. Find the cotangent of 87° 57' 59" 



Ans. 9.884592. 
Ans. 8.550356. 



CASE II. 

To find the degrees, minutes and seconds, answering to any 
given logarithmic sine, cosine, tangent or cotangent. 

35. Search in the table, and in the proper column, and if the 
number be found, the degrees will be shown either at the top 
or bottom o f the page, and the minutes in the side columns, 
either at the left or right. 

But, if the number cannot be found in the table, take 
from the table the degrees and minutes answering to the near- 
est less logarithm, the logarithm itself, and also the corres- 
ponding tabular difference. Subtract the logarithm taken from 
the table from the given logarithm, annex two ciphers to the 
remainder, and then divide the remainder by the tabular dif- 
ference : the quotient will be seconds, and is to be connected 
with the degrees and minutes before found ; to be added for 
the sine and tangent, and subtracted for the cosine and co- 
tangent. 

EXAMPLES. 

1. Find the arc answering to the sine 9.8 80054 
Sine 4d° 20', next less in the table 9.8799C3 

Tabular difference . . . 1.81)91.00(50" 

Hence, the arc 49° 20' 50" corresponds to the given sine 
9.880054. 

2. Find the arc whose cotangent is . 10.008688 
cot 44° 26', next less in the table . 1C.0C8591 

Tabular difference . . . 4.21)97.00(23" 



TRIGONOMETRY 



18U 



Thooroms. 



Hence, 44° 26' — 23" = 44° 25' 37" is the arc answering to 
the given cotangent 10.008088. 

3. Find the arc answering to tangent 9.979110. 

Ans. 43° 37' 21". 

4 Find the arc answering to cosine 9.944599. 

Ans. 28° 19' 45". 

36. We shall now demonstrate the principal theorems of 
Plane Trigonometry. 




THEOREM I. 

The sides of a plane triangle are proportional to the sines 
of their opposite angles. 

Let ABC be a triangle; then will 

CB : CA : : sin A : sin B. 
For, with A as a centre, and AD 
equal to the less side BC, as a radius, 
describe the arc DI: and with B as 
a centre and the equal radius BC, ^ EI L F 

describe the arc CL : now DE is the sine of the angle A, 
and CF is the sine of B, to the same radius AD or BC 
But by similar triangles, 

AD : DE : : AC : CF. 
But AD being equal to BC, we have 

BC : sin A : : AC : sin B, or 
BC : AC : : sin A : sin B. 
By comparing the sides AF AC, in a similar manner, we 
should find, AB : AC : : s\n C r sin 5. 



190 



TRIGONOMETRY. 



Theorems. 



THEOREM II. 



tan \{C-B). 




In any triangle, the sum of the two sides containing either 
angle, is to their difference, as the tangent of half the sum of 
the two other angles, to the tangent of half their difference. 
Let ACB be a triangle: then will 
AB + AC: AB-AC: : tan i(C + B) 

With A as a centre, and a radius 
AC the less of the two given sides, 
let the semicircle IFCE be de- 
scribed, meeting AB in /, and BA 
produced, in E. Then, BE will 
be the sum of the sides, and BI 
their difference. Draw CI and A F. 

Since CAE is an outward angle of the triangle ACB, it 
is equal to the sum of the inward angles C and B (Bk. 
I, Th. xvi.) But the angle CIE being at the circumference, 
is half the angle CAE at the centre (Bk. II, Th. viii. Cor- 
1) ; that is, half the sum of the angles C and B, or equal 
lo $(C+B). 

The angle AFC = ACB, is also equal to ABC + BAF ; 
therefore, BAF ' = ACB - ABC. 

But, ICF= i(BAF) = \(ACB — ABC), or \(C— B). 

With / and C as centres, and the common radius IC, let 
the arcs CD and IG be described, and draw the lines CE and 
IB perpendicular to IC. The perpendicular CE will pass 
through E, the extremity of the diameter IE, since the right 
angle ICE must be inscribed in a semicircle. 

But CE is the tangent of CIE = \{C+B); and IB is the 
tangent of ICB = %(C — B), to the common radius CI. 



TRIGONOMETRY. 



191 



Theorems, 



But since the lines CE and Iff are parallel, the triangles 
BUI and BCE are similar, and give the proportion, 

BE : BI :: CE : III, or 

by placing for BE and BI, CE and III, their values, we have 

AB + AC : AB— AC : : tan £(<?+£) : tan J(tf — B). 



THEOREM III, 

In any plane triangle, if a line is drawn from the vertical 
angle perpendicular to the base, dividing it into two segments: 
tlien, the whole base, or sum of the segments, is to the sum of 
the two other sides, as the difference of those sides to the dif- 
ference of the segments. 

Let BAC be a triangle, and AD perpendicular to the base; 
then will 

BC: CA + AB :: CA — AB: CD — DB 

For, ~A1? = ~Bl? +~AD* 

(Bk. IV, Th. xii) ; 

and AC* = IK? + ~KI) 

by subtraction AC — AB — CD — 
BlP. 

But since the difference of the squares 
of two lines is equivalent to the rectangle contained by their sum 
and difference (Davies' Legendre, Bk. IV, Prop, x,) we have, 

AC* — AB'=(AC+AB) . (AC—AB) 
ind W — ~DB % = (CD +DB).(CD — DB) 

therefore, (CD + DB) . (CD — DE) = (AC+AB) .(AC— A B) 
hence, CD -\- DB : AC 4- AB : -.AC—AB: CD — DB. 




192 



TRIGONOMETRY 

T h e o r e m a . 



THEOREM IV. 

In any right-angled plane triangle, radius is to the tan- 
gent of either of the acute angles, as the side adjacent to On 
tide opposite. 

Let CAB be the proposed triangle, 
and denote the radius by R : then will 
R: tan C::AC : AB. 
For, with any radius as CD describe ^" 
the arc DH, and^draw the tangent DG. 

From the similar triangles CDG and CAB we have 
CD :DG :: CA : AB; hence, 
R : tan C : : CA : A B. 
By describing an arc with B as a centre, we could show in 
the same manner that, 

R • tan B : : AB : AC. 




THEOREM V. 

In every right-angled plane triangle, radius is to the cosine 
of either, of the acute angles, as the hypothenuse to the side 
adjacent. 

Let ABC be a triangle, right-ano-led 
at B then will 

R : cos A : : A C : AB. 
For, from the point A as a centre, with - 
any radius as AD, describe the arc DF, 
which will measure the angle A, and draw DE perpendicular 
to AB : then will AE be the cosine of A. 

The triangles ADE and ACB, being similar, we have 
AD : A.E \\AC : AB : that is, 
R : cos A : : AC : AB. 




El? 



TRIGONOMETRY. 193 

Applications. 

Remark. The relations between the sides and angles of 
plane triangles, demonstrated in these five theorems, are suf- 
ficient to solve all the cases of Plane Trigonometry. Of the 
six parts which make up a plane triangle, three must be given, 
and at least one of these a side, before the others can be de- 
termined. 

If the three angles are given, it is plain, that an indefi- 
nite number of similar triangles may be constructed, the 
angles of which shall be respectively equal to the angles 
that are given, and therefore, the sides could not be de- 
termined. 

Assuming, with this restriction, any three parts of a trian« 
gle as given, one of the four following cases will always be pre 
scnted. 

I. When two angles and a side are given. 
II. When two sides and an opposite angle are given. 

III. When two sides and the included angle are given. 

IV. When the three sides are given. 

CASE L 

When two angles and a side are given. 

Add the given angles together and subtract their sum from 
ISO degrees. The remaining parts of the triangle can then 
bo found by Theorem I. 

EXAMPLES. 

1. In a plane triangle ABC, there 
are given the ang'e A = 58° 07', the 

angle B ^ 22° 37 , and the side AB = A^ 

108 *ards. Required the other parts. 
17 




194 TRIGONOMETRY. 

Applications. 

GEOMETRICALLY. 

Draw an indefinite straight line AB, and from the 8<-ale of 
equal parts lay off AB equal to 408. Then at A lay off an 
angle equal to 58° 07', and at B an angle equal to ?2° 37', 
and draw the lines AC and BC: then will ABC bo the tri- 
angle required. 

The angle C may be measured either with the protractor or 
the scale of chords (Arts. 16 and 1*7), and will bo found equal 
to 99° 16'. The sides AC and BC may be measured by re- 
ferring them to the scale of equal parts (Art. 2). We shall 
find AC = 158.9 and BC = 351. yards. 

TRIuONOMETRICALLY BY LOGARITHMS. 



To the angle . 


. A = 58° 07' 




Add the angle 


. B = 22° 37' 




Their sura 


= 80° 44' 




taken from . . 


180° 00' 




leaves C . . 


99° 16' which, 


exceeding 90" 


we use its supplement 


80° 44'. 




To 


find the side BC. 




As sin C 99° 16' 


ar. comp. . 


0.005705 


: sm A 58° 07' 


. 


9.928972 


: : AB 408 


. 


2.610660 


BC 351.024 


(after rejecting 10) 


2.545337 



Remark. The logarithm of the fourth term of a proportion 
is obtained by adding the logarithm of the second term to that 
of the third, and subtracting from their sum the logarithm of 
the first term. But to subtract the first term is the same as 



T R 1 G N O Si E T R Y. 195 



Applications 



to add its arithmetical complement and reject 1C from the sum 
(Art. 13) : hence, the arithmetical complement of the first 
term added to the logarithms of the second and third terms, 
minus ten, will give the logarithm of the fourth term. 





To fii 


id side AC. 




As sin C 


99° 16' 


ar. com p. 


0.005705 


: sin B 


22° 37' 


• 


9.584968 


: : AB 


408 


• • • 


2.610660 


: AC 


158.976 


• • • 


2.201333 



2. In a triangle ABC, there are given A = 38° 25', 
B = 57° 42', and AB = 400 : required the remaining parts 
Ans. C= 83° 53', £(7=249.974, AC = 340,04 



CASE II. 

When two sides and an opposite angle are given. 
In a plane triangle ABC, there are C 

given AC = 216, CB = 117, the 
angle A = 22° 37', to find the other . 
parts. 

GEOMETRICALLY. 

Draw an indefinite right line ABB': from any point as A y 
draw AC making BAC = 22° 37', and make AC = 216. 
With O as a centre, and a radius equal to 117, the other given 
side, describe the arc B'B ; draw B' C and BC: then will 
cither of the triangles ABC or AB' C, answer all the condi- 
tions of the question. 




196 TRIGONOMETRY. 



Applicati ins, 



TRIG ONOMETRIC ALLY. 

To find the angle B. 

As BO 117 . ar. comp. . . 7.931814 

: AC 216 2.334454 

: sin A 22° 37' . . > . 9.584 908 

: sin B' 45° 13' 55", or ABC 134° 4G' 05" 9.851236 

The ambiguity in this, and similar examples, arises in con 
sequence of the first proportion being true for either of the 
angles ABC, or AB'C, which are supplements of each other, 
and therefore have the same sine (Art. 30). As long as the 
two triangles exist, the ambiguity will continue. But if the 
side CB, opposite the given angle, is greater than AC, the arc 
BB' will cut the line ABB', on the same side of the point A, 
in but one point, and then there will be only one triangle an- 
swering the conditions. 

If the side CB is equal to the perpendicular Cd, the art 
BB' will be tangent to ABB', and in this case also there 
will be but one triangle. When CB is less than the perpen- 
dicular Cd, the arc BB' will not intersect the base ABB', and 
in that case, no triangle can be formed, or it will be impossible 
to fulfil the conditions of the problem. 

2. Given two sides of a triangle 50 and 40 respectively, and 
the angle opposite the latter equal to 32° : required the re- 
maining parts of the triangle. 

Aks. If the angle opposite the side 50 is acute, it is equal 
to 41° 28' 59" ; the third angle is then equal to 106° 31' 01", 
und the third side to 72.368. If the angle opposite the sidp 



TRIGONOMETRY. 19T 



Applications. 



50 is obtuse, it is equal to 138° 31' 01", the third- angle to 
9° 28' 59", and the remaining side to 12.436. 



case in. 
When the two sides and their included angle are given. 



B 



X, 



Let ABC be a triangle; AB } BC, 
the given sides, and B the given 
angle. 

Since B is known, we can find the 
sum of the two other angles : for A- 

A + C == 180° - B and 
l(A + C) = 1(180° - B) 
We next rind half tho difference of the angles A and C by 
Theorem ii., viz. 

BC + BA: BC- BA : : tan {(A + C) : tan \(A - C): 
in which we consider BC greater than BA, and therefore A ia 
greater than C\ since the greater angle must be opposite the 
greater side. 

Having found half the difference of A and C, by adding it 
to the half sum, ^(A -\- C), we obtain the greater angle, and by 
subtracting it from half the sum, we obtain the less. That b 
\{A + C) + \{A - C) = A, and 
\{A+ C)-i(A - C)= a 
Having found the angles ^1 and C, the third side AC may 
be found by the proportion. 

sin A : sin B : : BC : AC. 

EXAMPLE8. 

1. In the triangle ABC, Jet BC = 540, AB = 450, and 
the included angle B = 80° : required the remaining parts. 
17* 



198 TRIGONOMETRY. 



Application a. 



GEOMETRICALLY. 

Draw an indefinite right line BC and from any point, as 
B t lay off a distance BC = 540. At B make the angle 
CBA — 80°: draw BA and make the distance Bi -- 450; 
tlraw AC; then will ABO be the required triangle. 

TRIGONOMETRICALLY. 

BC + BA = 540 4- 450 = 990; and BC — BA = 540 — 

450 = 90. 

A + C 7 = 180° — 5 = 180° —80° = 100°, and therefore, 

J(,l + C) = 3(100°) = 50° 

To find \{A— C). 
As BC -\- BA 990 . ar. comp. . 7.004365 

BC — BA 90 ... 1.954243 

: tan ](A + C) 50° ... 10.076187 

tan 1(^1— 6') 6° 11' . . . 9.03 4795 

Hence, 50° 4- 0° 11' = 5G° 11' == A; and 50° — 0° 11' == 
43° 49' = C. 

To find the third side AC. 

As sin C 43° 49' . ar. comp. . . 0.159672 

: sin B 80° 9.993351 

:: vlS 450 2.653213 

AC 640X82 2.806230 

2o Given two sides of a plane triangle, 1686 and 960, and 
their included angle 128° 04': required the other parts. 

Ans. Angles, 33° 34' 39" ; 18° 21' 21" ; side 2400. 



TRIGONOMETRY. 



199 



Applications. 



CASE IV. 

Having given the three sides of a plane triangle, to find 
the angles. 

Let fall a perpendicular from the angle opposite the greater 
tide, dividing the given triangle into two right-angled triangles : 
then find the difference of the segments of the base by Theo- 
rem iii. Half this difference being added to half the base 
gives the greater segment ; and, being subtracted from half the 
base, gives the less segment. Then, since the greater segment 
belongs to the right-angled triangle having the greatest hypo- 
thenuse, we have the sides and right angle of two right-angled 
triangles, to find the acute angles. 




EXAMPLES. 

1. The sides of a plane trian- 
gle being given; viz. BC = 40, AC 
= 34 and AB — 25 : required the 
angles. H 

GEOMETRICALLY. 

With the three given lines as sides construct a triangle as 
in Bk. II. Prob. xi. Then measure the angles of the triangle 
either with the protractor or scale of chords. 

TRIGONOMETRIC ALLY. 

As BC :AC + AB : : AC - AB : CD - BD 

That is, 40 : 59 : : 9 : 59 X 9 = 13.275 

40 
40 -f- 13.275 



Then, 



40 
= 26.6375 = CD 



. , 40 - 13.275 

AJid = 13.3625 = BD. 



200 TRIGONOMETRY 



Applications. 



In the triangle DAC, to find the angle DAC. 

As AC 34 . . ar. corap. . 8.468521 

DC 26.6375 .... 1.425493 

sin D 90° 10.000000 

sin DAC 51° 34' 40" . . . 9.8940U 



In the triangle BAD. to find the angle BAD. 
As AB 25 ar. cornp. . 8.602060 

BD 13.3625 . . . 1.125887 

sin D 90° ... 10.000000 

sin BAD 32° 18' 35" . . . 9.727947 
Hence 90° — D A C = 90° — 51° 34' 40" = 38° 25' 20" = C 
and 90° — BAD = 90° — 32° 18' 35" = 57° 41' 25" = B 
and BAD + DAC = 51° 34' 40" -f 32° 18' 35" = 83° 53' 
15" = A. 

2. In a triangle, in which the sides are 4, 5 and 6, what are 
the angles ? 

Ars. 41° 24' 35"; 55° 46' 16" ; and 82° 49' 09". 

SOLUTION OP RIGHT-ANGLED TRIANGLES. 

The unknown parts of a right-angled triangle may be found 
by either of the four last cases : or, if two of the sides are 
given, by means of the property that the square of the hypo- 
thenuee is equivalent to the sura of the squares of the two other 
sides. Or the parts may be found by Theorems iv. and v. 

EXAMPLES. 

1. In a right-angled triangle BAC, 
there are given the hypothenuse BC 
— 250, and the base AC = 240: re- C 
quired the other parts. 




TRIGONOMETRY. 201 



Applications. 




To find the angle B. 




As BC 


250 . ar com p. 


7.602060 


: AC 


240 ... 


2.380211 


: : sin A 


90° ... 


10.000000 


: sin B 


73° 44' 23" . 


9.982271 



But C = 00° — B = 90° — 73° 44' 23" = 16° 15' 37" : 

Or C may be found from the proportion. 

A& CB 250 ar. comp. . 7.602060 

2.380211 
10.000000 



CB 


250 


AC 


240 


R 


. 


C 


16° 15' 37' 



cos C 16° 15' 37" . . . 9.982271 



To find side AB by Theorem Iv. 

As R ar. comp. . 0.000000 

tan C 16° 15' 37" . . . 9.404889 

AC 240 .... 2.3S0211 

AB 70.0003 .... 1.845100 



2. In a right- angled triangle BAC, there are given AC ~ 
384, and B = 53° 08' : required the remaining parts. 

Ans. AB= 287.96; £(7= 479.979; (7= 36° 52'. 

DEFINITIONS. 

1. A horizontal angle is one whose sides are horizontal ; its 
plane is also horizontal. 

2. An angle of elevation or depression, has one horizontal side, 
said the other oblique, but lying directly above or below the first. 



202 



TRIGONOMETRY 



Applications. 



APPLICATION TO HEIGHTS AMD DISTANCES. 



PROBLEM I. 

To determine the horizontal distance to a point which is inac- 
cessible by reason of an intervening river. 

Let <J be the point. Measure 
along the bank of the river s hori- 
zontal base line AB, and select the 
stations A and B, in such a manner 
that each can be seen from the other, 
and the point C from both of them. 
Then measure the horizontal angles 
CAB and CBA y with an instrument adapted to that purpose. 

Let us suppose that we have found AB = 600 yards. 
CAB = 57° 35' and CBA = 64° 51'. 




The angle C = 180° — {A + B) = 57° 34' 

To find the distance BC. 



As 


sin C 


57° 34' ar. comp. 


0.073649 


i 


sin A 


57° 35' ... 


9.926431 


: : 


AB 


GOO .... 


2.778151 




BC 


600.11 yards. 
To find the distance AC, 


2.778231 


As 


sin 


57° 34' ar. comp. 


0.073649 




siD B 


64° 51' ... 


9.956744 


; : 


AB 


600 . ; 


2.778151 




AC 


043.94 yards. . 


2.808544 



TE IQONOMETRY, 



103 



Applications. 







PROBLEM II. 

To determine the altitude of an inaccessible object above a 

given horizontal plane. 

FIRST METHOD 

Suppose D to bo the inaccessible 
object, and BC the horizontal plane 
from which the altitude is to be 
estimated: then, if we suppose DC 
to be a vertical line, it will repre- 
sent the required distance. 

Measure any horizontal base line, as BA ; and at the ex- 
tremities B and A, measure the horizontal angles CBA and 
CAB. Measure also, the angle of elevation DBC. 

Then in the triangle CBA there will be known, two anglea 
and the side AB ; the side BC can therefore be determined. 
Having found BC, we shall have, in the right-angled triangle 
DBC, the base BC and the angle at the base, to find the per- 
pendicular DC, which measures the altitude of the point D 
above the horizontal plane BC. 

Let us suppose that we have found 
BA - 780 yards, the horizontal angle CBA = 41° 24', 
the horizontal angle CAB = 96° 28', and the angle of eleva- 
tion DBC= 10° 43'. 

In the triangle BCA, to rind the horizontal distance BC. 

The angle BCA=1 80° - (41° 24' -f 96° 28') = 42° 08'= C 

As sin C . 42° 08' ar. comp. . 0.113369 

sin A . 96° 28' . . . . 9.997228 

AB . 780 .... 2.892095 

BC . 1155.29 .... 3.062692 



204 



TRIGONOM ETRY 



Applications. 




In the right-ano-led triangle DEC, to find DC. 



As 



R 


ar. coin p. 


0.000000 


DEC 


10° 43' 


9.2 77 04 3 


BC 


1155.29 


3.0G2C9 2 


DC 


218.64 


2.339735 



tan 



Remark I. It might, at first, appear that the solution which 
we have given, requires that the points B and A should be in 
the same horizontal plane ; but it is entirely independent of 
such a supposition. 

For, the horizontal distance, which is represented by BA, 
i« the same, whether the station A is on the same level with 
B, above it, or below it. The horizontal angles CAB and 
CEA are also the same, so long as the point C is in the verti- 
cal line DC. Therefore, if the horizontal line through A should 
cut the vertical line DC, at any point as E, above or below C, 
AB would still be the horizontal distance between B and A, 
and AE which is equal to AC, would be the horizontal dis- 
tance between A and C. 

If at A, we measure the angle of elevation of the point D 
we shall know in the right-angled triangle DAE, the base AE, 
and the angle at the base; from which the perpendicular I) E 
can be determined. 



TRIGONOMETRY. 205 



k Applications. 



Let us suppose that we had measured the angle of elevation 
DAE, and found it equal to 20° 15'. 

Firs': In the triangle BAC, to find AC or its equal AE. 

As sin 42° 08' ar. comp. . 0.1733G9 

i sin B 41° 24' ... 9.820406 

11 AB 780 ... 2.892095 

AC 768.9 . . . 2.885870 



In the right angled triangle DAE, to find DE. 

A3 R ar. comp. . . 0.000000 

: tan A 20° 15' . . . 9.5GG932 

: : AE 768.9 . . . _ 2.885870 

DE 283.66 . . . 2.452802 

Now, since DC is less than DE, it follows that the station 
B is above the station A. That is, 

DE — DC= 283.66- 218.64 = 65.02 = EC, 
which expresses the vertical distance that the station B if* 
above the station A. 

Kemark II. Tt should be remembered, that the vertical dis 
tance which is obtained by the calculation, is estimated from 
a horizontal line passing through the eve at the time of ob- 
servation. Hence, the height of the instrument is to be added, 
in order to obtain the true result 

8ECOND METHOD. 

When the nature of the ground will admit of it, measure a 
base line AB in the direction of the object D. Then mea- 
sure with the instrument the angles of elevation at A and B. 

Then, since the outward angle DBC is equal to the sum 
18 



206 



TRIGONOMETRY. 



Applications 




of the angles A and 
ABB, it follows, that 
the angle ABB is 
equal to the difference 
of the angles of ele- 
vation at A and B. Hence, we can find all the parts of the 
triangle ABB. Having found BB y and knowing the angle 
DBC, we can find the altitude BC. 

This method supposes that the stations A and B are on 
the same horizontal plane ; and therefore can only be used 
when the line AB is nearly horizontal. 

Let us suppose that we have measured the base line, and 
the two angles' of elevation, and 

(AB = 975 yards, 
A ■= 15° 36', 
BBC = 27° 29'; 
required the altitude BC. 

First: ABB = BBC- A = 27° 29' - 15° 36' = 11° 53' 



In the triangle ABB, to find BD. 
As sin B 11° 53' ar. comp. 

sin A 15° 30' ... 

AB 975 ... 

BB 1273.3 



0.686302 
9.429623 

2.989005 
3.104930 



In the triangle BBC, to find BC. 

as R ar. comp. . 0.000000 

sin B 27° 29' ... 9.664163 

BB 1273.3 . . . 3.104930 

BC 587.61 . . . 2.769093 



TRIGONOMETRY, 



207 



Applications. 




5%= 5=- 




PROBLEM III. 

To determine the perpendicular distance of an object below a 
given horizontal p>lane. 

Suppose C to be directly over 
the given object, and A the point 
through which the horizontal plane 
is supposed to pass. 

Measure a horizontal base line ^^/J^^^^^^^^^^g 

AB, and at the stations A and B SfeS^SP 
conceive the two horizon taF lines 

AC, BC\ to be drawn. The oblique 
lines from A and B to the object will be the hypothenusea 
of two right-angled triangles, of which AC, BC, are the 
bases. The perpendiculars of these triangles will be the dis- 
tances from the horizontal lines AC, BC, to the object. If 
we turn the triangles about their bases AC, BC, until they 
become horizontal, the object, in the first case, will fall at C", 
and in the second at C". 

Measure the horizontal angles CAB, CBA, and also thfl 
angles of depression C AC, C" BC. 

Let us suppose that we have 

AB = 672 yards 
BAC --= 72° 29' 
ABC = 39° 20' 
C'AC=21° 49' 
. C"BC = 19° 10' 

First: In the triangle ABC, the horizontal angle ACS =? 
80° - (A 4- B) = 180° - 111° 49' = 68° 11'. 



found 



208 



TRIGONOMETRY 



Applications. 










As 



Aft 



To find tbe horizontal distance AC. 

sin C G8° 11' ar. comp. . 0.032275 

sin B 39° 20' ... 9.801973 

AB 672 ... 2.827369 

AC 45S.79 . . . 2.661017 

To find the horizontal distance BC. 

sin C 68° 11' . ar. comp. . ..0.032275 

sin A 72° 29' .... 9.979380 

AB 672 2.827369 

BC 690.28 2.839024 



As 



As 



In the triangle CAC' y to find CC. 

R . ar. comp. . . 0.000000 

tan C'AC 27° 49' .... 9.722315 

AC 458.79 .... 2.661617 

CC 242.06 .... 2.383932 

In the triangle CBC", to find CC" 

R . ar. comp. . . 0.000000 

tan C'BC 19° 10' . . . 9.541061 

BC 690.28 .... 2.839024 

CC" 239.93 . , 2.380083 



TRIGONOMETRY 



209 



Applications, 



Hence also, CC - CC" = 242.06 - 239.93 = 2.13 yard?, 
which is the height of the station A above station B. 

PROBLEMS. 

1. Wanting to know the distance between two inaccessible 
objects, which lie in a direct line from the bottom of a tower 
of 120 feet in height, the angles of depression are measured, 
and are found to be, of the nearer 57°, of the more remote 
25° 30' : required the distance between them. 

Ans. 113.656 feet. 

2. In order to find the distance between 
two trees A and B, which could not be 
directly measured because of a pool which 
occupied the intermediate space, the dis- 
tances of a third point C from each of 
them wore measured, and also the included 
angle A CB : it was found that 

CB — 612 yards 
CA = 588 yards 
ACB = 55° 40'; 
required the distance AB. 

Ans. 592.907 yards. 

3. Being on a horizontal plane, and wanting to ascertain 
the height of a tower, standing on the top of an inaccessible 
bill, there were measured, the angle of elevation of the top 
of the hill 40°, and of the top of the tower 51°; then mea- 
suring in a direct line 180 feet farther from the hill, the angle 
of elevation of the top of the tower was 33° 45'; required tl.e 
height of the towei. 

Ans. 83.998 feet. 
18* 




210 



TRIGONOMETRY, 



Application 



4. Wanting to know the horizon- 
tal distance between two inaccessi- 
ble objects E and W y the following 
measurements were made, 

f AB = 536 yards 
BAW — 40° 16' 
WAE = 57° 40' 
ABE = 42° 22' 
EBW= 71° 07' 
required the distance EW. 




A 



Ans. 939.634 yards. 




5. Wanting to know the 
horizontal distance between two 
inaccessible objects A and B, 
and not finding any station 
from which both of them could 
be seen, two points C and D, 
were chosen, at a distance from 
each other, equal to 200 yards ; from the former of these points 
A could be seen, and from the latter B, and at each of the 
points C and D a staff was set up. From C a distance CF 
was measured, not in the direction DC y equal to 200 yards, 
and from T> a distance DE equal to 200 yards, and the follow- 
ing angle* taken, 

r AFC = 83° 00' BDE = 54° 30' 
viz. } ACD = 53° 30' BBC = 156° 25' 

[ ACF= 54° 31' BED = 88° 30' 

Ans. AB = 345.467 yards. 



APPLICATIONS 



GEOMETRY. 



MENSURATION OF SURFACES. 
DEFINITIONS. 

1 The area of any figure has already been defined to be 
the measure of its surface (Bk. IV. Def. 7). This measure is 
merely the number of squares which the figure contains. 

A square whose side is one inch, one foot, or one yard, 
&c, is called the measuring unitj and the area or contents of 
a figure is expressed by the number of such squares which 
the figure contains. 

2. In the questions involving decimals, the decimals are 
generally carried to four places, and then taken to the nearest 
figure. That is, if the fifth decimal figure is 5, or greater 
than 5, the fourth figure is increased by one. 

3. Surveyors, in measuring land, generally use a chain 
called Gunter's chain. This chain is four rods, or 66 feet in 
length, and is divided into 100 links. 

4. An acre is a surface equal in extent to 10 square chains; 
that is, equal to a rectangle of which one side is ten chains 
and the other side one chain. 

One quarter of an acre, is called a ~ood. 

Since the chain is 4 rods in length, 1 square cha'n contains 
16 square rods; and therefore, an acre, which is 10 square 
chains, contains 160 square rods, and a rood contains 40 
square rods. The square rods are called perches. 



212 APPLICATIONS 

Mensuration of Surfaces. 

5. Land is generally computed in acres, roods, and perched 
which are respectively designated by the letters A, R, P. 

When the linear dimensions of a survey are chains or linke 
the area will be expressed in square chains or square links, 
and it is necessary to form a rule for reducing this area to 
acres, roods, and perches. For this purpose, let us form ibe 
following 

TABLE. 

1 square chain = 1 00 x 1 00 = 1 0000 square links 
1 acre = 10 square chains = 100000 square links 

1 acre = 4 roods = 1 GO perches. 
I square mile = 6400 square chains = 640 acres. 

6. Now, when the linear dimensions are links, tho area 
will be expressed in square links, and may be reduced to 
acres by dividing by 100000, the number of square links in an 
acre : that is, by pointing off five decimal places from the 
right hand. 

If the decimal part be then multiplied by 4, and five places 
of decimals pointed off from the right hand, the figures to the 
left hand will express the roods. 

If the decimal part of this result be now multiplied by 40, 
and five places for decimals pointed off, as before, the figures 
to the left will express the perches. 

If one of the dimensions be in links, and the other in chains, 
the chains may be reduced to links by annexing two ciphers, 
or, the multiplication may be made without annexing the ci- 
phers, and the product reduced to acres and decimals of an 
acre, by pointing off three decimal places at the right hand. 

When both dimensions are in chains, the product is re- 



OF GEOMETRY. 213 

Mensuration of Surfaces. 

luced to acres by dividing by 1 0, or pointing off one decimal 
place. 

From which we conclude : that, 

I. If Imks be multiplied by links, the product is reduced to 
acres by pointing off five decimal places from the right hand. 

II. If chains be multiplied by links, the product is reduced to 
acres by pointing off three decimal places from the right hand. 

III. If chains be multiplied by chains, the product is reduced 
to acres by pointing off one decimal place from the right hand. 

7. Since there are 16.5 feet in a rod, a square rod is equaj 
to 16.5 x 16.5=272.25 square feet. 

If the last number be multiplied by 160, we shall have 

272.25 X 160 = 43560 the square feet in an acre. 
Since there are 9 square feet in a square yard, if the last 
number be divided by 9, we obtain 

4 840 = the number of square yards in an acre 

problem I. 

To find the area of a square, a rectangle, a rhombus, or a 
parallelogram. 

RULE. 

Multiply the base by the perpendicular height and the produc- 
will be the area (Bk. IV. Th. viii). 

EXAMPLES. 



I Required the area of the square 
A BCD, each of whose sides is 36 feet 



D C 


1 


A B 



214 APPLICATIONS 



Mensuration of Surfaces 



We multiply two sides of 
the square together, and the 
product is the area in square 
feet. 



Operation. 
36x36=1296 sq. ft. 



2. How many acres, roods, and perches, in a square whose 
side is 35.25 chains? Arts. 124 A. 1 R. 1 P 

3. What is the area of a square whose side is 8 feet 4 
inches? Ans. 69 ft 5' 4" 

4. What is the contents of a square field whose side is 46 
rods? Ans. 13 A. R. 36 P. 

5. What is the area of a square whose side is 4769 yards 1 

Ans. 22743361 sq. yds 



6. What is the area of the parallelo- 
gram ABCD, of which the base AB is 
64 feet, and altitude DE, 36 feet ? 



e i 



c 

7 



We multiply the base 64, 
by the perpendicular height 
36, and the product is the re- 
quired area. 



Operation 
64x36 = 2304 sq ft 



7. What is the area of a parallelogram whose base is 12,25 
yards, and altitude 8.5? Ans 104,125 sq. yds. 

8. What is the area of a parallelogram whose base is 8.75 
chains, and altitude 6 chains ? Ans. 5 A. 1 R OP. 

9. What is the area of a parallelogram whose base is 7 r eot 
9 inches, and altitude 3 feet 6 inches ? 

Ans. 27 sq.ft. 1 ' 6' 



OF GEOMETRY 



215 



Mensuration of Surfaceo 



10. To find the area of a rectangle 
A BCD, of which the base AB = A5 
yards, and the altitude AD = 15 yards. 

Here we simply multiply 
the base by the altitude, and 
the product is the area. 



B 

Operation 

45xl5=z675 sq. yds. 



1 1. What is the area of a rectangle whose base is 14 feot 
6 inches, and breadth 4 feet 9 inches ? 

Ans. 68 sq.ft. 10' 6". 

12. Find the area of a rectangular board whose length is 
112 feet, and breadth 9 inches. Ans. 84 sq. ft. 

13. Required the area of a rhombus whose base is 10.51 
and breadth 4.28 chains. Ans. 4 A. 1 R. 39.7 P+. 

14. Required the area of a rectangle whose base is 12 feot 
6 inches, and altitude 9 feet 3 inches. 

Ans. 115 sq. ft. T 6" 

PROBLEM II. 

To find the area of a triangle, whea the base and altitude 
are known. 

RULE. 

I. Multiply the base by the altitude, and half the product will 
be the area. 

II. Multiply the base by half the altitude and the product inill 
be the area (Bk. IV. Th. ix). 

EXAMPLES. 

I Required the area of the trianglo 
ABC, whose base AB is 10,75 feet, 
and altitude 7,25 feet. 

15 




216 APPLICATIONS 



Mensuration of Su rfaces. 

Operation. 
We first multiply the base 10,75x7,25=77,9375 

by the altitude, and then di- and 

vide the product by 2. 77,9375^2 = 38,96875 

= area 

2. What is the area of a triangle whose base is 18 feet 4 
inches, and altitude 11 feet 10 inches? 

Ans. 108 sq. ft. 5' 8". 

3. What is the area of a triangle whose base is 12.25 
chains, and altitude 8.5 chains? Ans. 5 A. OR. 33 1\ 

4. What is the area of a triangle whose base is 20 feet, 
and altitude 10.25 feet. Ans. 102.5 sq. ft. 

5. Find the area of a triangle whose base is 625 and alti- 
tude 520 feet. Ans. 162500 sq. ft 

6. Find the number of square yards in a triangle whose 
base is 40 and altitude 30 feet. Ans. 66^ sq. yds. 

7. What is the area of a triangle whose base is 72.7 yards, 
and altitude 36.5 yards ? Ans. 1326,775 sq. yds 

problem ill. 
To find the area of a triangle when the three sides are 
known. 

RULE, 

I. Add the three sides together and take half their sum. 

II. From this half sum take each side separately. 

III. Multiply together the half sum and each of the three 
remainders, and then extract the square root of the product, 
which will be the required area. 



OF GEOMETRY. 217 

Mensuration of Surface.?-. 
EXAMPLES. 

1. Find the area of a triangle whose sides are 20, 30, and 
10 rods. 

20 45 45 45 

30 20 30 40 



_5_^ 25 1st rem. 15 2d rem. 5 3d rem 

45 half sum, 

Then, to obtain the pruduct, we have 

45x25x 15x5 = 84375; 
from which we find 



area^v 784375 ^' 290 ' 4737 perches. 

2. How many square yards of plastering are there in a tri- 
angle, whose sides are 30, 40, and 50 feet ? Ans. 66§. 

3. The sides of a triangular field are 49 chains, 50.25 
chains, and 25.69 : what is its area ? 

Ans. 61 A. 1 R. 39,68 P 

4. What is the area of an isosceles triangle, whose base is 
20, and each of the equal sides 15 ? Ans. Ill 803. 

5. How many acres are there in a triangle whose three 
sides are 380, 420 and 765 yards. Ans. 9 A. OR. 38 P. 

6. How many square yards in a triangle whose sides are 
13, 14, and 15 feet. Ans. 9j. 

7 What is the area of an equilateral triangle whose side 
is 25 feet ? Ans. 270.6329 sq. ft. 

8. What is the ares of a triangle whose sides are 24, 36, 
and 48 yards? Ans 418.282 sq. yds. 



2i8 



APPLICATIONS 



Mensuration of Surfaces. 



PROBLEM IV. 

To find the hypothenuse of a right angled triangle when 
the base and perpendicular are known 

RULE. 

I. Square each of the sides separately. 

II. Add the squares together. 

III. Extract the square root of the sum, which will be ifa Inj- 
pothenuse of the triangle (Bk. IV. Tli. xii). 

EXAMPLES. 

1. In the right angled triangle ABC, 
we have, AB = 30 feet, BC = 40 feet, to 
find tIC. 



We first square each side, 
and then take the sum, of 
which we extract the square 
root, which gives 




3(T 



900 



40 =1600 



sum =2500 



^4 0=^/2500 = 50 feet. 

2. The wall of a building, on the brink of a river, is 120 
feet high, and the breadth of the river 70 yards : what is the 
length of a line which would reach from the top of the wall to 
the opposite edge of the river? Arts. 241.86 ft. 

3. The side roofs of a house of which the eaves are of the 
same height, form a right angle at the top. Now, the length 
of the rafters on one side is 10 feet, and on the other 14 feet : 
what is the breadth of the house ? Arts. 17.204 ft. 

4. Wh it would be the width of the house, in the last ex« 
ample, if the rafters on each side were 10 feet? 

Ans. 14.142 ft. 



OF GEOMETRY. 



219 



Mensuration of Surfaces 



5. What would be the width, if the rafters on each side 
were 14 feet ? Ans. 19.7989 ft. 

PROBLEM V. 

When the hypothenuse and one side af a right angled tri- 
angle are know n, to find the other side 

RULE. 

Square the hypothenuse and also the other given side, and 
take their difference : extract the square root of this difference, 
and the result will be the required side (Bk. IV. Th. xii. Cor.). 

EXAMPLES. 

1. In the right angled triangle ABC, 
there are given 

AC=50 feet, and AB — 40 feet, 
required the side BC. 

We first square the hypoth- 
enuse and the other side, after 
which we take the difference, 
and then extract the square 
root, which gives 

BC=*/900=z30 feet. 

2 The height of a precipice on the brink of a river is 103 
feet, and a line of 320 feet in length will just reach from the 
top of it to the opposite bank : required the breadth of the 
river. Ans. 302.9703 ft. 

3. The hypothenuse of a triangle is 53 yards, and the per 
pendicular 45 yards : what is the base 1 Ans. 28 yds. 

4 A ladder 60 feet in length, wil] reach to a window 40 




Operation 

50* = 2500 

40 2 rrl600 

Diff.= 900 



220 APPLICATIONS 



Mensuration of Surfaee 



feet from the ground on one side of the street, and by turning 
it over tu the other side, it will reach a window 50 feet from 
the ground : required the breadth of the street. 

Ans. 77.8875 ft. 

PROBLEM VI. 

To find the area of a trapezoid. 

RULE. 

Multiply the sum of the parallel sides by the perpendicular 
distance between them, and then divide the product by two : the 
quotient will be the ana (Bk. IV. Th. x). 

EXAMPLES. 



1 . Required the area of the trapezoid 
A BCD, having given 



E B 
,45-321.51 feet, DC = 214.24 fret, and OE = 171.16 feei 

Operation. 



We first find the sum of the 
sides, and then multiply it by 
the perpendicular height, after 
which, we divide the product 
by 2, for the area. 



321.51+214.24 = 535.75- 
sum of parallel sides. 

Then, 
535.75x171.16 = 91698.97 
91698.97 
2 
i =the area. 



2 What is the area of a trapezoid, the parallel sides of 
which, are 12.41 and 8.22 chains and the perpendicular dis- 
tance between them 5.15 chains ? 

Ans. 5 A. 1 R. 9.956 P. 

3. Required the area of a trapezoid whose parallel sides 



OF GEOMETRY. 221 

Mensuration of Surfaces. 

are 25 feet 6 inches, and 18 feet 9 inches, and the perpen- 
dicular distance between them 10 feet and 5 inches. 

Ans. 230 sq. ft. 5' 7". 

4. Required the area of a trapezoid whose parallel sides 
are 20.5 and 12.25, and the perpendicular distance between 
them 10.75 yards. Ans. 176.03125 sq. yds. 

5. What is the area of a trapezoid whose parallel sides are 
7.50 chains, and 12.25 chains, and the perpendicular height 
15.40 chains? Arts. 15 A. R. 33.2 P 

PROBLEM VII. 

To find the area of a quadrilateral. 



Measure the four sides of the quadrilateral, and also one of the 
diagonals : the quadrilateral will thus be divided into two trian- 
gles, in both of which all the sides will be known. Then, find 
the areas of the triangles separately, and their sum will be the 
area of the quadrilateral. 

EXAMPLES. 



1. Suppose that we have meas- 
ured the sides and diagonal A C, of 
the quadrilateral ABCD, and found 




A 



A 5 = 40.05 chains; CD =29.87 chains, 
50=26.27 chains AD = 37.01 chains, 

and A C = 55 chains : 

required the area of the quadrilateral 

Ans. 101 A. 1 R 15 P 
19* 



122 



a PPLICATIONS 



Mensi rat 



of Surfaces 




Remark. — Instead of measuring 
the four sides of the quadrilateral, 
we may let fall the perpendicu- 
lars Bb, Dg, on the diagonal AC. 
The area of the triangles may then 
be determined by measuring these 
perpendiculars and diagonal AC. The pendiculars arv,Dg — 
18.95 chains, and BL=z]7.92 chains. 

2. Required the area of a quadrilateral whose diagonal ifi 
HO. 5, and two perpendiculars 24.5, and 30.1 feet. 

Arts. 2197.65 sq.ft. 

3. What is the area of a quadrilateral whose diagonal is 
108 feet 6 inches, and the perpendiculars 56 feet 3 inches, 
and 60 feet 9 inches ? Ans. 6347 sq.ft. 3'. 

4. How many square yards of paving in a quadrilateral 
whose diagonal is 65 feet, and the two perpendiculars 28, and 
33| ^et ? Ans. 222 T V sq. yds. 

5. Required the area of a quadrilateral whose diagonal is 
42 feet, and the two perpendiculars 18, and 16 feet. 

Ans. 714 sq. ft. 

6. What is the area of a quadrilateral in which the diago- 
nal is 320.75 chains, and the two perpendiculars 69.73 chains, 
and 130.27 chains ? Ans. 3207 A. 2 R. 



PROBLEM VIII. 

To find the area of a regular polygon. 



Multiply half the perimeter of the figure by the perpendicular 
Lei fall from the centre on one of the sides, and *lte product will 
he the area (Bk. IV. Th. xxvi) 



OP GEOMETRY. 



223 



Mensuration of Surfaces. 



EXAMPLES. 



1. Required the area of the regular 
pentagon ABCDE, each of whose -® 1 
sides AB, BC, &c, is 25 feet, and 
the perpendicular OP, 17.2 feet. 



We first multiply one side 
by the number of sides and 
divide the product by 2 : this 
gives half the perimeter which 
we multiply by the perpen- 
dicular for the area. 




25x5 



: 62.5 = half the perim- 



eter. Then, 

625x17.2 = 1075 sq. /*.=the 

area. 



2. The side of a regular pentagon is 20 yards, and the per- 
pendicular from the centre on one of the sides 13,76382 ; re- 
quired the area. 

A'ns. 688.191 sq. yds. 

3. The side of a regular hexagon is 14, and the perpen- 
dicular from the centre on one of the sides 12.1243556: re- 
quired the area. 

Ans. 509.2229352 sq.ft. 

4. Required the area of a regular hexagon whose side is 
14.6, and perpendicular from the centre 12.64 feet. 

Ans. 553.632 sq ft. 

5. Required the area of a heptagon whose side is 19,3" 

ar.d perpendicular 20 feet. 

Ans. 1356.6 sq. ft. 

The following table shows the areas of the ten regular 



;24 



A PPLICATiONS 



Mensuration of Surfaces 



polygons when the side of each is equal to 1 : it also shows 
the length of the radius of the inscribed circle. 



Number of 

6ides. 


Names. 


Areas. 


Radius of inscribed] 
circle. 


3 


Triangle, 


0.4330127 


0.2886751 


4 


Square, 


1.0000000 


0.5000000 


5 


Pentagon, 


1.7204774 


0.6881910 


6 


Hexagon, 


2.5980762 


0.8660254 


7 


Heptagon, 


3.6339124 


1.0382617 


8 


Octagon, 


4.8284271 


1.2071068 


9 


Nonagon, 


6.1818242 


1.3737387 


10 


Decagon, 


7.6942088 


1.5388418 


11 


Undecagon, 


9.3656404 


1.2028437 


12 


Dodecagon, 


11.1961524 


1.8660254 



Now, since the areas of similar polygons are to each othei 
as the squares described on their homologous sides (Bk. IV 
Th. xx), we have 

l 3 : tabular area : : any side squared : area. 

Hence, to find the area of a regular polygon, we have the 
following 

RULE. 

I Square the side of the polygon. 

II. Multiply the square so found, by the tabular area set oppo- 
site the polygon of the same number of sides, and the product 
will be the irea. 

EXAMPLES. 

1 . What is the area of a regular hexagon whose side is 20 



20 =400 



and tabular area =2,5980762. 



Hence, 



2.5980762 x 400 = 1039.23048 = the area. 



OFGEOMETRV. 225 



Mensuration of Surfaces 



2. What is the area of a pentagon whose side is 25 ? 

Ans. 1075.298375. 

3. What is the area of a heptagon whose side is 30 feet 

Ans. 3270.52116 

4. What is the area of an octagon whose side is 10 feet \ 

Ans. 482.84271 sq. ft 
b. The side of a nonagon is 50 : what is its area ? 

Ans. 15454.5605 

6. The side of an undecagon is 20 : what is its area ? 

Ans. 3746.25616. 

7. The side of a dodecagon is 40 : what is its area ? 



Ans. 17913.84384 



PROBLEM IX. 



To find the area of a long and irregular figure, bounded on 
one side by a straight line. 

RULE. 

I. Divide the right line or base into any number of equal 
parts, and measure the breadth of the figure at the points of di 
vision, and also at the extremities of the base. 

II. Add together the intermediate breadths, and half the sum 
of the extreme ones- 
Ill. Multiply this sum by the base line, and divide the product 

bif ite number of equal parts of the base. 

EXAMPLES. 

1. The breadths of an irregu- a 

lar figure, at five equidistant a r \^^\ \ \ 

places, A, B, C, D, and E, be- j_. i — ^ — -4 ^ 

ing 8.20 chains, 7.40 chains. 



220 APPLICATIONS 

Mensuration of Surfaces. 

9.20 chains, 10.20 chains, and 8.60 chains, and the whole 

length 40 chains : required the area. 

8.20 35.20 

8.60 40 

2 )16\80 4)1408.00 

8.40 mean of the extremes. 352.00 square chains. 
7.40 
9.20 
1O.20 
35.20 the sum. 

Ans. 35 A. 32 P. 

2. The length of an irregular piece of land being 21 chains 
and the breadths, at six equidistant points, being 4.35 chains 
5.15 chains, 3.55 chains, 4.12 chains, 5.02 chains, and 6.10 
chains : required the area. Ans. 9 A. 2 R. 30 P. 

3. The length of an irregular figure is 84 yards, and the 
breadths at six equidistant places are 17.4 ; 20.6 ; 14.2 ; 16.5; 
20.1 ; and 24.4 : what is the area ? Ans. 1550.64 sq. yds. 

4. The length of an irregular field is 39 rods, and its 
breadths at five equidistant places, are 4.8; 5.2; 4.1; 7.3, 
and 7.2 rods : what is its area ? Ans. 220.35 sq. rods. 

5. The length of an irregular field is 50 yards, and its 
breadths at seven equidistant points, are 5.5 ; 6.2 ; 7.3 ; 6 ; 
7.5 ; 7 ; and 8.8 yards : what is its area ? 

Ans. 342.916 sq. yds. 

6. The length of an irregular figure being 37.6, and the 
breadths at nine equidistant places, 0; 4.4 ; 6.5 ; 7,6 ; 5.4 ; 8; 
5.2 ; 6.5 ; and 6.1 : what is the area? Ans. 219.255. 

PROBLEM X. 

To find the circumference of a circle when the diameter is 
known. 



OF GEOMETRY. 227 

Mensuration of Surfaces. __ 

RULE 

Multiply the diameter by 3.1416, and the product will be th& 
circumference. 

EXAMPLES. 

1. What is the circumference of a circle whose diameter 
is 17? 



We simply multiply the 
number 3.141G by the diam- 
eter and the product is the 
circumference 



Operation. 
3.1416x17 = 53.4072, 
which is the circumference. 



2. What is the circumference of a circle whose diameter ie 
40 feet? Ans. 125.664 ft. 

3. What is the circumference of a circle whose diameter is 
12 feet? Ans. 37.6992 ft. 

4. What is the circumference of a circle whose diameter is 
22 yards? Ans. 69.1152 yds. 

5. What is the circumference of the earth — the mean diam- 
eter being about 7921 miles? Ans. 24884.6136 mi. 

PROBLEM XI. 

To find the diameter of a circle when the circumference is 
known. 

RULE. 

Divide the circumference by the number 3.1416 and the quo- 
tient unll be the diameter. 

EXAMPLES. 

1. The circumference of a circle is 69.1152 yards: what 
is the diameter' 



228 APPLICATIONS 



Mensuration 


of Surfaces. 


We simply divide the cir- 
cumference fey 3.1416, and 
the quotient 22 is the diam- 
eter sought. 


Operation. 

3.1416)691152(22 
62832 
62832 
62832 



2. What is the diameter of a circle whose circumference is 
11652.1944 feet ? Ans. 3709. 

3. What is the diameter of a circle whose circumference is 
6850? Ans. 2180.4176. 

4. What is the diameter of a circle whose circumference is 
50? Ans. 15.915. 

5. If the circumference of a circle is 25000.8528, what is 
the diameter ? Ans. 7958. 

PROBLEM XII. 

To find the length of a circular arc, when the number ot 
degrees which it contains, and the radius of the circle are 
known. 

RULE. 

Multiply the number of degrees by the decimal .01745, and 
the product arising by the radius of the circle. 

EXAMPLES. 

1 . What is the length of an arc of 30 degrees, in a circle 
whose radius is 9 feet. 

We merely multiply the Operation. 

given decimal by the number .01 745 x 30 x 9 = 4.71 1 5, 
of degrees, an 1 by the radius, which is the length of the arc 

Remark. — When the arc contains degrees and minutes, re- 
duce the minutes to the decimals of a degree, which is done 
by dividing them by 60. 



OF GEOMETRY. 



229 



Mensuration of Surfaces 



2. What is the length of an arc containing 12° 10' oj 
\2i° the diameter of the circle being 20 yards ? 

A as. 2.1231 

3. "What is the length of an arc of 10° 15' or 10j°, in a 
circle "whose diameter is 68? Ans. 6.0813. 



PROBLEM XIII. 

To find the length of the arc of a circle when the chord 
dad radius are given. 

RULE. 

1. Find the chord of half the arc. 

1 T From eight times the chord of half the arc, subtract the 
chord of the whole arc, and divide the remainder by 3, and the 
quotient will be the length of the arc, nearly. 

EXAMPLES. 

1. The chord AB = 30 feet, and the 
radius AC =20 feet: what is the 
length of the arc ADB ? 

First draw CD perpendicular to the 
chord AB : it will bisect the chord at 
P, and the arc of the chord at D. 
Then AP =15 feet. Hence, 

AC 2 -AP*=CP l : that is, 
400—225 = 175 and V 175=1 3.228 = CP 
Then OD-CP=20-13.228 = 6.772=Z>P. 




Again, 
hencej 
Then, 



^D = yjlP 2 +PD 2 = y'225 + 45.859984 
AD = 16.4578 = chord of the half arc. 

l6 -± 578 * 8 - 3 ° = 33.8874 = arc ADB. 



20 



230 APPLICATIONS 



Mensuration of Surfac 



2 What is the length of an arc the chord of which is 24 
feet, and the radius of the circle 20 feet ? 

Ans. 25.7309 jt. 

3. The chord of an arc is 16 and the diameter of the circle 
20 : what is the length of the arc ? Ans. 18.5178. 

4. The chord of an arc is 50, and the chord of half the 
arc is 27 : what is the length of the arc 1 Ans. 55 $. 

PROBLEM XiV. 

To find the area of a circle when the diameter and circum- 
ference are both known. 

RULE. 

Multiply the circumference by half the radius and the product 
will be the area (Bk. IV. Th. xxvii). 

EXAMPLES. 

1. What is the area of a circle whose diameter is 10, and 
circumference 31.416 ? 

If the diameter be 10, the 
radius is 5, and half the ra- 
dius is 2| : hence, the cir- 
cumference multiplied by 2^ 
gives the area. 

2. Find the area of a circle whose diameter is 7; and cir- 
cumference 21.9912 yards. Ans. 38.4846 yds. 

3. How many square yards in a circle whose diameter is 
3£ feet, and circumference 10.9956. Ans. 1.069016. 

4. What is the area of a circle whose diameter is 100, and 
circumference 314.16? Ans. 7854 



Operation. 

31.416 x 2^=78.54; 
which is the area. 



OF GEOMETRY. 23 1 



Mensuration of Surfaces 



5. What is the area of a circle whose diameter is I , and 
circumference 3.1416. Ans. 0.7854. 

6. What is the area of a circle whose diameter is 40, and 
circumference 131.9472 ? Ans. 1319.472. 

PROBLEM XV. 

To find the area of a circle when the diameter only Is 
known. 

RULE. 

Square the diameter, and then multiply by the decimal .7854 

EXAMPLES. 

What is the area of a circle whose diameter is 5 ? 



We square the diameter, 
which gives us 25, and we 
then multiply this number 
and the decimal .7854 to- 
gether. 



Operation. 

.7854 

5j=_25 

39270 

15708 

area= 19.6350' 



2. What is the area of a circle whose diameter is 7 ? 

Ans. 38.4846. 

3. What is the area of a circle whose diameter is 4,5 ? 

Ans. 15.90435. 

4. What is the number of square yards in a circle whose 
diameter is 1 1 yards ? Ans. 1.069016. 

5. What is the area of a circle whose diameter is 8.75 
feet? Ans. 60.1322 sq.ft. 

PROBLEM XVI. 

To find the area of a circle when the circumference only 
is known. 



232 APPLICATIONS 



Mensuration of Surface3 



RULE. 

Multiply tlie square of the circumference by the decimal .07958, 
and the product will be the area very nearly 

EXAMPLES. 

1. What is the area of a circle whose circumference ie 
3.1416? 



We first square the cir- 
cumference, and then multi- 
ply by the decimal .07958. 



Operation. 

)65 
,07958 



3.1416 2 =9,86965056 



area =.7854 + 



2. What is the area of a circle whose circumference is 9H 

Ans. 659.00198. 

3. Suppose a wheel turns twice in tracking 16-2 feet, and 
that it turns just 200 times in going round a circular bowling- 
green : what is the area in acres, roods, and perches ? 

Ans. 4 A. 3 R. 35.8 T 

4. How many square feet are there in a circle whose cir 
cumference is 10.9956 yards? Ans. 86.5933. 

5. How many perches are there in a circle whose circuin 
ference is 7 miles ? Ans. 399300.608. 

PROBLEM XVII. 

Having given a circle, to find a square which shall have an 
squal area. 

RULE. 

I. The, diameter X. 8862 = side of an equivalent square 

II. TJw circumference X .2821= side of an equivalent square 



OF GEOMETRY. 233 



Mensuration of Surfaces 



EXAMPLES. 

1. The diameter of a circle is 100 : what is the side of a 
square of equal area ? Ans. 88.62. 

2. The diameter of a circular fishpond is 20 feet, what 
would be the side of a square fishpond of an equal area? 

Ans. 17.724 ft. 

3. A man has a circular meadow of which the diameter is 
875 yards, and wishes to exchange it for a square one of equal 
size : what must be the side of the square ? 

Ans. 775.425. 

4. The circumference of a circle is 200 : what is the side 
of a square of an equal area 1 Ans. 56.42. 

5. The circumference of a round fishpond is 400 yards : 
what is the side of a square pond of equal area 1 

Ans. 112.84. 

6. The circumference of a circular bowling-green is 412 
yards : what is the side of a square one of equal area ? 

Ans. 116.2252 yds. 

7. The circumference of a circular walk is 625 : what is 
the side of a square containing the same area ? 

Ans. 176.3125. 

PROBLEM XVIII. 

Having given the diameter or circumference of a circle, to 
find the side of the inscribed square. 

RULE. 

I. The diameter X .7071 =zside of the inscribed square. 

II. The circumference X .2251 —side of the inscribed square. 

20* 



231 



APPLICATIONS 



Mensuration of Surfaces, 



EXAMPLES. 



1. Tlio diameter AB of a circle 
is 400 : what is the value of AC, 
the side of the inscribed square 1 

Here, 

.7071 x 400=282.8400= AC. 




2. The diameter of a circle is 412 feet: what is the side 
of the inscribed square? Ans. 291.3252 ft. 

3. If the diameter of a circle be 600 what is the side of 
the inscribed square ? Ans 424.26. 

4. The circumference of a circle is 312 feet: what is the 
side of the inscribed square ? Ans. 70.2312 ft. 

5. The circumference of a circle is 819 yards : what is the 
side of the inscribed square 1 Ans. 184.3569 yds. 

6. The circumference of a circle is 715 : what is the side 
of the inscribed square ? Ans. J 60.9465. 

7. The circumference of a circular walk is 625 : what is 
ihe side of an inscribed square 1 Ans. 140.6875. 



PROBLEM XIX 

To find the area of a circular sector. 

RULE. 

I. Find the length of the arc by Problem XII. 

II. Multiply the arc by one half the radius, and the product 
will be the area 



OF GEOMETRY. 



23 s 



Mensuration of Surfac 



EXAMPLES. 




1. What is the area of the circular 

sector ACB, the arc AB containing 

18°, and the radius CA being equal to 
3 feet. 



First, .0T1745X IS x3 = . '94230 = length AB. 
Then, ,94230x1 J=l. 4 1345= area 

2. What is the area of a sector of a circle in which the ra- 
dius is 20 and the arc one of 22 degrees ? 

Ans. 76.7800. 

3. Required the area of a sector whose radius is 25 and 
the arc of 147° 29'. Ans, 804.2448. 

4. Required the area of a semicircle in which the radius is 
13. Ans. 265.4143. 

5. What is the area of a circular sector when the length of 
the arc is 650 feet and the radius 325 ? 

Ans. 105625 sq. ft. 

PROBLEM XX. 

To find the area of a segment of a circle. 

RULE. 

I. Find the area of the sector having the same arc with ihs. 
segment, by the last Problem. 

II. Find the area of the triangle formed by the chord of the 
segment and the two radii through its extremities. 

Ill If the segment is greater than the semicircle, add the two 
areas together; but if it is less, subtract them, and the result in 
eithei case, will be the area required 



236 



A PPLICATIONS 



Mensuration of Surf 



EXAMPLES. 



1. What is the area of the seg- 
ment ADB, the chord AB = 24 
feet and CA =20 feet. 



First, CP=^/CA 9 - 



AP 4 



= ^400 — 144 = li 
Then, 
PZ>=OD-OP = 20 — 16 = 4. 




And, AD=^/AP i +PD i =zy/T44-{- 16 = 12,04911 



then, 



arc ^B = 12 ' 6491 ' X8 - 8 i =8S ,7309. 



Arc ADB =25,7309 

half radius = 10 



area sector AD£C = 257,3090 
area 04£=192 



AP= 

CP= 

area CAB = 



1~92 



65,309 = area of segment ADB 



2. Find the area of the segment 
AFB\ knowing the following lines, 
viz: .45 = 20.5; FP= 17.17; AF 
—20; FG=ll-5; and 04 = 11.64. 



. .__ ^Gx8-i4F 11.5x8-20 o 
Arc AGF= = =24 : 

o o 

and sector AGFBC=24x 11.64=279.36 : 

but CP—FP— 40=17.17— 11.64=5.53: 

,^„ .4£xCP 20.5x5.53 
Then, area ACB= = = 56.6825 




O * GEOMETRY 



237 



Mensuration of Surfaces. 



Then, area of sector ^4F5C = 279.36 

do. of triangle ABC — 56.6825 
gives area of segment AFB = 336.0425 

3 What is the area of a segment; the radius of the circle 
being 10 and the chord of the arc 12 yards ? 

Ans. 16.324 sq. yds. 

4. Required the area of the segment of a circle whose 
chord is 16, and the diameter oi the circle 20. 

Ans. 44.5903. 

5. What is the area of a segment whose arc is a quadrant, 
the diameter of the circle being 18 ? Ans. 63.6174. 

6. The diameter of a circle is 100, and the chord of the 
seoment 60 : what is the area of the segment ? 

A /is. 408, nearly. 



PROBLEM XXI. 

To find the area of an ellipse. 

Multiply the two axes together, and their product by the decimal 

,7854, and the result will be the required area. 

EXAMPLES. 

1. Required the area of an ellipse, 
whose transverse axis AB = !0 feet, 
and the conjugate axis DE = 50 feet. 

^Bx£E = 70x50 = 3500: 

Then, ,7854x3500=2748.9 = area. 

2. Required the area of an ellipse whose axes are 24 and 
1 \ Ans. 339.2928. 




•238 APPLICATIONS 



Mensuration of Surfaces 



3. What is the area of an ellipse whose axes are 80 and 
CO ? Ans. 3769.92. 

4. What is the area of an ellipse whose axes are 50 and 
45? Ans. 1767.15. 

PROBLEM XXII. 

To find tlie area of a circular ring : that is, the area in- 
cluded between the circumferences of two circles, having a 
common centre. 

RULE. 

I. Square the diameter of each ring, and subtract *he square 
of the less from that of the greater. 

II. Multiply the difference of the squares by thb d ximai 
7854, and the product will be the area. 

EXAMPLES. 



1. In the concentric circles 
having the common centre C, w T e 

have 

A! 
AS =10 yds., and DE = 6 yards : 

what is the area of the space in- 
cluded between them ? 



BA* = 10 2 = \00 
DE*= ?= 36 




Difference = 64 
Then, 63 X. 7854 = 50.2656 =area. 

2. What is the area of the ring when the diameters of the 
circle aie 20 and 10 1 Ans. 235.62. 



OF GEOMETRY. 23<J 



Mensuration of Solid 



3. If the diameters are 20 and 15, what will be the area in- 
cluded between the circumferences ? Ans. 137.445. 

4. If the diameters are 16 and 10, what will be the area in- 
cluded between the circumferences 1 Ans. 122.5224. 

5 Two diameters are 21.75 and 9.5 ; required the area ol 
the circular ring. Ans. 300.6609 

6. If the two diameters are 4 and 6, what is the area of the 
ring? Ans. 15.708 



MENSURATION OF SOLIDS. 

DEFINITIONS. 

The mensuration of solids is divided into two parts. 
1st, The mensuration of the surfaces of solids : and 
2d, The mensuration of their solidities. 

We have already seen that the unit of measure for plane 
surfaces, is a square whose side is the unit of length (Bk. IV 
Def. 7). 

2. A curve line which is expressed by numbers is also re- 
ferred to an unit of length, and its numerical value is the num- 
ber of times which the line contains the unit. 

If then, we suppose the linear unit to be reduced to a 
straight line, and a square constructed on this* line, this square 
will be the unit of measure for curved surfaces. 

3. The unit of solidity is a cube, whose edge is the unit in 
which the linear dimensions of the solid are expressed ; and 



240 



APPLICATIONS 



Mensuration of Solids, 



the face of this cube is the superficial unit in which the but 
face of the solid is estimated (Bk. VI. Th. xiii. Sch). 

4 The following is a table of solid measure. 

1 cubic foot =1728 cubic inches. 
1 cubic yard = 27 
1 cubic rod = 4492^ 
1 ale gallon =282 
1 wine gallon =231 



I bushel 



cubic feet, 
cubic feet, 
cubic inches, 
cubic inches. 

= 2150,42 cubic inches. 



PROBLEM i. 
To find the surface of a right prism. 

RULE. 

Multiply the perimeter of the base by the altitude and the pro- 
duct will be the convex surface : and to this add the area of the 
bases, when the entire surface is required (Bk. VI. Th. i). 

EXAMPLES 



1. Find the entire surface of the 
regular prism whose base is the reg- 
ular polygon ABODE and altitude 
AF, when each side of the base is 
20 feet and the altitude AF, 50 feet. 




D 



B C 

AB+BC+CD+DE + EA = 100; and .4^=50 : then 
(AB + BC+ CD+DE + EA) x AF= convex surface 



OF GEOMETRY. 24j 

Mensuration of S o 1 i tl 3 . 



which becomes, 100x50 = 5000 square feet ; which is the 
convex surface. For the area of the end, we have 
AB x tabular number = area ABODE, 
drat is, 20 2 x tabular number, or 400 X 1.720477 = 088. ! 1)06 = 
the area ABODE. 

Then, eoiiv«x surface = 5000 square feet, 

lowei base 688.1908 square feet, 

upper base 088.1908 square feet. 

Entire surface G37G.381G 



2. What is the surface of a cube, the length of each siae 
being 20 feet? Ans. 2400 sq. ft. 

3. Find the entire surface of a triangular prism, whose base 
is an equilateral triangle, having each of its sides equal to 18 
inches, and altitude 20 feet. Ans. 91.949 sq. ft. 

4. What is the convex sirface of a regular octagonal prism, 
'he side of whose base is 15 and altitude 12 feet ? 

Ans. 1440 sq. ft. 

5. What must be paid for lining a rectangular cistern v\ th 
lead -at 2d a pound, the thickness of the lead being such as to 
require lib. for each square foot of surface ; the inner dimen- 
sions of the cistern being as follows : viz. the length 3 feet 2 
inches, the breadth 2 feet 8 inches, and the depth 2 feet G 
inches? Ans. £2 3s I0|^r. 

PROBLEM 11 

To find the solidity of a prism. 

RULE. 

Multiply tkt area of the base by the perpendicular height, and 
the product will be the solidity. 



212 



APPLICATIONS 



Mensurat'on of Soldi 



EX UIPLES. 



1 . What is the solidity of a reg- 
ular pentagonal prism whose altitude 
is 20, and each side of the base 15 
feet ? 

To find the area of the base we 
have by Problem VIII. page 178. 




15 2 -225: and 225x1.7204774 = 387.107415 = 

the area of the base : hence, 

387.10741 5 X 20 = 7742.1483 = solidity. 

2. What is the solid contents of a cube whose side is 24 
inches ? Ans. 13824 solid in. 

3. How many cubic feet in a block of marble, of which the 
length is 3 feet 2 inches, breadth 2 feet 8 inches, and height 
or thickness 2 feet 6 inches 1 Ans. 21 £ solid ft. 

4. How many gallons of water, ale measure, will a cistern 
contain whose dimensions are the same as in the last ex- 
ample ? Ans. 129H- 

5. Required the solidity of a triangular prism t? nose alti- 
tude is 10 feet, and the three sides of its triangular base 3, 4, 
and 5 feet. Ans. 60 solid ft. 

G. What is the solidity of a square prism whoee height is 
5J feet, and each side of the base \\ fcot? 

Ans 91 solid ft. 



OF GEOMETRY 



243 



Mensuration of Solids. 



7. What is the solidity of a prism whose base is an equi- 
lateral triangle, each si-de of which is 4 feet, the height of the 
prism being 10 feet? Ans. G9.282 solid ft. 

8 What is the number of cubic or solid feet in a regular 
pentagonal prism of which the altitude is 15 feet and each 
Ride of the base 3.75 feet ? Ans. 362.913 

PROBLEM III. 

To find the surface of a regular pyramid. 

RULE. 

Multiply the perimeter of the base by half the slant lipight, 
and the product will be the convex surface : to this add the area 
of the base, if the entire surface is required (Bk. VI. Th vi) 



EXAMPLES. 

1. In the regular pentagonal pyramid 
S— ABODE, the slant height SF is 
equal to 45, and each side of the base 
is 15 feet: required the convex sur- 
face, and also the entire surface. 

15 X 5 =z75 = perimeter of the base, 
75x22^ = 1687.5 square feet = area of 
convex surface. 



And 15 =225: then 225 X 1.7204774 = 387.1074 15 = the area 

of the base. 

Hence, convex surface =1687.5 

area of the base= 387.107415 
Entire surface =2074.607415 square feet. 




244 



APPLICATIONS 



Mensuration of Solids. 



2. What is the convex surface of a regular triangular pyra- 
mid, the slant height being 20 feet, and each side of tho ba3o 
3 feet 1 Ans. 90 sq. ft 

3. What is the entire surface of a regular pyramid whose 
blant height is 15 feet, and the base a regular pentagon, of 
which each side is 25 feet? ' Ans. 2012.798 sq. ft 



PROBLEM IV. 

To find the convex surface of the fiustum of a regulai 
pyramid. 

RULE. 

Multiply half the sum of the perimeters of the two bases by 
the slant height of the frustum, and the product will be the con - 
vex surface (Bk. VI. Th. vii). 

EXAMPLES. 

1. In the frustum of the regular pen- 
tagonal pyramid each side of the lower 
base is 30, and each side of the upper 
base is 20 feet, and the slant height 
f'F is equal to 15 feet. What is the 
convex surface of the frustum ? 

Ans. 1875 sq. ft. 

2. How many square feet are there in the convex surface 
of ihe frustum of a square pyramid, whose slant height is 10 
feet, each side of the lower base 3 feet 4 inches, and each 
side of the upper base 2 feet 2 nches ' Ans. 110. 

3. What is the convex surface oi the frustum of a heptago 

nal pyramid whose slant height is 55 feet, each side of the 

lowei base 8 feet, and each side of the upper base 4 feet ? . 

Ans. 2310 sq. ft 




OF GEOMETRY 



245 



Mensuration o 1 Solids. 
PROBLEM V. 

To find the solidity of a pyramid. 

RUl E. 

Multiply the area of the base by the altitude and divide the pro- 
duct by 3, the quotient will be the solidity (Bk. VI. Th. xvii), 

FXA.MP1.ES. 



1 What is ilie solidity of a pyramid 
the area of whose base is 215 sqttare 
feet and the altitude SO— 45 feet ? 

First, 215x45 = 9f>75: 

then, 9675- 3=3225 
which is the solidity expressed in solid 
feet. 




2. Required the solidity of a square pyramid, each side of 
its base being 30 and its altitude 25. Ans. 7500 solid ft. 

3. How many solid yards are there in a triangular pyramid 
whose altitude is 90 feet, and each side of its base 3 yards? 

Ans. 38.97117. 

4. How many solid feet in a triangular pyramid the altitude 
;)f which is 14 feet 6 inches, and the three sides of its base 5, 
6 and 7 feet? Ans. 71.0352. 

5. What is the solidity of a regular pentagonal pyramid, its 

altitude being 12 feet, and each side of its base 2 feet ! 

Ans 27-527G solid ft. 
21* 



2iC) 



APPLICATIONS 



Mensuration of Solids 



6 How many solid feet in a regular hexagonal pyramid 
whose altitude is G.4 feet, and each side of the base 6 inches' 

.4ns. 1.38504. 

7. How many solid feet are contained in a hexagonal pyra- 
mid the height of which is 45 feet, and each side of the base 
10 feet? . Ans. 3897.1113. 

8. The spire of a church is an octagonal pyramid, each sido 
of the base being 5 feet 10 inches, and its perpendicular 
height 45 feet. Within is a cavity, or hollow part, each side 
of the base being 4 feet 11 inches, and its perpendiculai 
height 41 feet: how many yards of stone does the spire 
contain 9 Ans. 32.197353 

PROBLEM VI. 

To imd the solidity of the frustum of a pyramid. 

RULE. 

Add together the areas of the tico bases of the frustum and 
a geometrical mean proportional between them ; and then multi- 
ply the sum by the altitude, and take one-third the product for 
the solidity. 

EXAMPLES. 



1. What is the solidity of the frus- 
tum of a pentagonal pyramid the area 
of the lower base being 16 and of the 
upper base 9 square r eet, the altitude 



being 7 feet ? 




OF GEOMETRY. 24T 



Mensuration of Solida. 



First, 16x9 = 144: then, -/l 44= 12, the mean 

Then, area of lower base =16 

area of upper base = 9 

mean of bases* =12 

_ 37 

height 7 

3) 209 

solidity = 86^ solid ft. 

2. What is the number of solid feet in a piece of timbei 
whose bases are squares, each side of the lower base being 
15 inches, and each side of the upper base being 6 inches, 
the length being 24 feet ? Ans. 19.5. 

3. Required the solidity of a regular pentagonal frustum, 
whose altitude is 5 feet, each side of the lower base 18 
mchcs, and each side of the upper base 6 inches. 

Ans. 9.31925 solid ft. 

4. What is the contents of a regular hexagonal frustum, 
whose height is 6 feet, the side of the greater end 18 inches, 
and of the less end 12 inches ? Ans. 24.681724 cubic ft. 

5. How many cubic feet in a square piece of timber, the 
areas of the two ends being 504 and 372 inches, and its 
length 31| feet? Ans. 95.447. 

6. What is the solidity of a squared piece of timber, its 
length being 18 feet, each side of the greater base 18 inches, 
and each side of the smaller 12 inches ? 

Ans. 28.5 cubic ft. 

7. What is the solidity of the frustum of a regular hexago- 
nal pyramid, the side of the greater end being 3 feet, that of 
the less 2 feet, and the height 12 feet? 

Ans. 197.453776 solid ft 



!48 



APPLICATIONS 



Mensuration of So lido. 



MEASURES OF THE THREE ROUND BODIES 
PROBLEM I 

To find the surface of a cylinder. 



Multiply the circumference of the base by the altitude, and the 
product will be the convex surface ; and to this, add the areas of 
the two bases, when the entire surface is required (Bk. VI. Th. ii) 



EXAMPLES. 

1. What is the entire surface of the 
cylinder in which AB, the diameter of 
the base, is 12 feet, and the altitude EF 
30 feet ? 

First, to find the circumference of the 
base, (Prob. X. page 180) : we have 
3.1416 x 12 = 37.6992 = circumference of 
the base. 

Then, 37.6992 x 30= 1 130.9760 = convex surface. 

Also, 12 2 =144: and 144 x .7854= 1 13.0976 = area of the 

base. 

Then. convex surface =1130.9760 

lower base 1 13.0976 

upper base 1 13.0976 

Entire area =1357.1712 




2. What is the convex curface of a cylinder, the diameter 
of whose base is 20, and the altitude 50 feet ? 

Arts 3141.6*7./*. 



OF GEOMETRY 



249 



Mensuration of the Round Bodies. 

3. Required the entire surface of a cylinder, whose altitude 
is 20 feel and the diameter of the base 2 feet. 

Ans. 131.9472 ft. 

4. What is the convex surface of a cylinder, the diametei 
ol whose base is 30 inches, and altitude 5 feet? 

Ans. 5654.88 sq. in. 

5. Required the convex surface of a cylinder, whose alti- 
tude is 14 feet, and the circumference of the base 8 feet 4 
inches. Ans. 116.6666, &c, sq. ft. 

PROBLEM n. 
To find the solidity of a cylinder. 

RULE. 

Multiply the area of the base by the altitude, and the prodw 
will be the solidity. 



EXAMPLES. 

1. What is the solidity of a cylinder, 
the diameter of whose base is 40 feet, 
and altitude EF, 25 feet ? 

First, to find the area of the base, we 
have (Prob. xv. page 231). 

40 2 =1600: then, 1 600 x .7854=1256.64. 

=area of the base. 

Then, 1256.64 x 25=31416 solid feet, which is the solidity. 

2. What is the solidity of a cylinder, the diameter of whose 
base is 30 feet, and altitude 50 feet ? 

Ans 35343 cubic ft. 




250 APPLICATIONS 



Mensuration of the Round Bodie 



6. What is the solidity ol a cylinder whose height is 5 feet, 
and the diameter of the end 2 feet? Ans. 15.708 solid ft. 

4. What is the solidity of a cylinder whose height is 20 
feet, and the circumference of the base 20 feet ? 

Ans. 636.64 cubic ft 

5. The circumference of the base of a cylinder is 20 feet, 
and the altitude 19.318 feet: what is the solidity? 

Ans. 614.93 cubic ft. 

6. What is the solidity of a cylinder whose altitude ie» 12 
feet, and the diameter of its base 15 feet ? 

Ans. 2120.58 cubic ft. 

7. Required the solidity of a cylinder whose altitude is 20 
feet, and the circumference of whose base is 5 feet 6 inches ? 

Ans. 48.1459 cubic ft. 

8. What is the solidity of a cylinder, the circumference of 
whose base is 38 feet, and altitude 25 feet ? 

Ans. 2872.838 cubic ft. 

9. What is the solidity of a cylinder, the circumference of 
whose base is 40 feet, and altitude 30 feet ? 

10. The diameter of the base of a cylinder is 84 yards, and 
the altitude 21 feet: how many solid or cubic yards does it 
contain ? Ans. 38792.4768. 

PROBLEM III. 

To find the surface of a cone. 

RULE. 

Multiply the circumference of the base by the slant height, and 
divide the product by 2 ; the quotient will be the convex surface, 
to which add the area of the base, when the entire surface is 
require i (Bk. VI. Th viii) 



OF GEOMETRY. 



251 



Mensuration of the Round Bodies. 



EXAMPLES. 




1. What is the convex surface of the 
cone whose vertex is C, the diameter 
AD, of its base being 8^ feet, and the 
side CA, 50 feet. 



First, 3.1416 X 81=26.7036 = circumference of base 

Then 2 -^L 6 2^ = G«7. 5 9 = convex surface. 
2 

2. Required the entire surface of a cone whose side is 36 
and the diameter of its base 18 feet. 

Ans. 1272.348 sq. ft. 

3. The diameter of the base is 3 feet, and the slant height 
J 5 feet : what is the convex surface of the cone ? 

Ans. 70.686 sq. ft. 

4. The diameter of the base of a cone is 4,5 feet, and* the 
slant height 20 feet : what is the entire surface ? 

Ans. 157.27635 sq. ft. 

5. The circumference of the base of a cone is 10. " 7 5. and 
the slant height is 18.25 : what is the entire surface ? 

Ans. 107.29021 sq. ft 

PROBLEM IV. 

To find the solidity of a cone. 

RULE. 

Multiply the area of the base by the altitude; and divide the pro- 
duct by 3, the quotient will be the solidity (Bk. VI. Th. ^viii). 



252 



APPLICATIONS 



Mensuration of the Round Bodies. 



EXAMPLES. 



1. What is the solidity of a cone, the 
area of whose base is 380' square feet, 
and altitude CB, 48 feet ? 




We simply multiply the 
area of the base by the alti- 
tude, and then divide the pro- 
duct by 3. 



Operation. 

380 

48 



3040 
1520 
3)18240 
area = 6080 



2. Required the solidity of a cone whose altitude is 27 
feet, and the diameter of the base 1 feet. 

Ans. 706.86 cubic ft. 

3. Required the solidity of a cone whose altitude is 1 0^ 
fpet, and the circumference of its base 9 feet ? 

Ans. 22.5609 cubic fl. 

4. What is the solidity of a cone, the diameter of whose 
base is 18 inches, and altitude 15 feet? 

Ans. 8.83575 cubic ft. 

5 The circumference of the base of a cone is 40 feet, and 
the altitude 50 feet: what is the solidity? 

Ans. 2122.1333 solid /?. 



OF GEOMETRY. 



253 



Mensuration of the Round Bodies 



PROBLEM V. 

To Jind the surface of the frustum of a cone 

RULE. 

Add together the circumferences of the twe bases; and multi- 
ply the sum by half the slant height of the frustum ; the product 
will be the convex surface, to which add the areas of the bases- 
when the entire surface is required (Bk. VI. Th. ix). 

EXAMPLES. 



. I. What is the convex surface of the 
frustum of a cone, of which the slant 
height is 12| feet, and the circumfe- 
rences of the bases 8,4 and 6 feet. 




half side 



Operation. 
8.4 
6 
14~4 
6.25 



We merely take the sum 

of the circumferences of the 

bases, and multiply by half 

the slant height, or side. 

area = 90 sq. ft. 

2. What is the entire surface of the frustum of a cone, the 
side being 16 feet, and the radii of the bases 2 and 3 feet ? 

Ans. 292.1688 sq.ft. 

3. What is the convex surface of the frustum of a cone, 
the circumference of the greater base being 30 feet, and of 
die less 10 feet; the slant height being 20 feet? 

Ans. 400 sq. ft. 

4. Required the entire surface of the frustum of a cone 

whose slant height is 20 feet, and the diameters of the bases 

8 and 4 feet Ans. 439.824 sq. ft 

22 



254 APPLICATIONS 



Mensuration of the Round B o d i e i 



PROBLEM VI. 

To find the solidity of the frustum of a cone 

RULE. 

I. Add together the areas of the two ends and a geometneal 
mean between them. 

II? Multiply this sum by one-third of the altitude and the 
product will be the solidity. 

EXAMPLES. 

1. How many cubic feet in the frus- 
tum of a cone whose altitude is 26 feet, M 
and the diameters of the bases 22 and II l 
1 8 feet 1 m//l/li 

First, 22 2 X.7854:=380.134 = area of Uk 
lower base : 
and 18 2 X .7854 = 254.47 = area of upper base. 



Then, ^380.134 X 254.47 = 31 1.0 18= mean. 

26 
Then, (380.134 + 254,47 + 31 1.018) x- =8195.39 which 

is life solidity. 

2. How many cubic feet in a piece of round timber the di- 
ameter of the greater end being 18 inches, and that of the less 
9 inches, and the length 14.25 feet ? Ans. 14.68943. 

3. What is the solidity of a frustum, the altitude being 1 8. 
the diameter of the lower base 8, and of the upper 4 ? 

Ans. 527.7888. 

4. If a cask, which is composed of two equal conic frus- 
tums joined together at their larger bases, have its bung di- 
ameter 28 inches, the head diameter 20 inches, and the length 



OF GEOMETRY. 



255 



Mensuration of the Round Bodies. 



40 inches, how many gallons of wine will it contain, there 
being 231 cubic inches in a gallon ? Ans. 79.0613. 

PROBLEM Vil. 

To find the surface of a sphere. 

RULE. 

Multiply the circumference of a great circle by the diameter, and 
the product will be the surface (Bk. VI. Th. xxiii). 

EXAMPLES. 



1. What is the surface of the sphere 
whose centre is C, the diameter being 
7 feet ? 

Ans. 153.9384 sq. ft. 



2. What is the surface of a sphere whose diameter is 24 ? 

Ans. 1809.5616. 

3. Required the surface of a sphere whose diameter is 
7921 miles. Ans. 19711 1024 sq. miles. 

4. What is the surface of a sphere the circumference o( 
whose great circle is 78.54? Ans. 1963.5. 

5. What is the surface of a sphere whose diameter is 1 3 
feet ? Ans. 5.58506 sq. ft. 

PROBLEM VIII. 

To find the convex surface of a spherical zone. 

RULE. 

Multiply the height of the zone by the circumference of a great 
circle of the sphere, and the product will be the convex surface 
(Bk. VI. Th. xxiv) 




256 



APPLICATIONS 



Mensuration of the Round Bodie 




EXAMPLES. 



1. What is the convex surface of 
the zone ABD, the height BE being 
9 inches, and the diameter of the 
sphere 42 inches ? 



First, 42x3.1416=131 .9472 = circumference, 

height = 9 

surface =1187.5248 square inches. 

2. The diameter of a sphere is 12| feet : what will be 
»he surface of a zone whose altitude is 2 feet ? 

Ans. 78.54 sq. ft. 

3. The diameter of a sphere is 21 inches : what is the sur- 
face of a zone whose height is \\ inches ? 

Ans. 296.8812 sq. in. 

4. The diameter of a sphere is 25 feet and the height of 
the zone 4 feet : what is the surface of the zone ? 

Ans. 314.16 sq. ft. 

5. The diameter of a sphere is 9, and the height of a zone 
3 feet : what is the surface oi the zone ? 

Ans. 84.8232. 

PROBLEM IX. 

To find the solidity of a sphere. 

rule :. 

Multiply the surface by one-third of the radius and the product 
will be the solidity (Bk. VI. Th. xxv)- 



OF GEO AI ETRY. 



251 



Mensuration of the Round B o 1 i 3 s . 




EXAMFLES 

1. What is the solidity of a sphere 
whose diameter is 12 feet? 

First, 3.1416x12—37.6992 = 

circumference of sphere. 

diameter = 12 

surface =452.3904 

one-third radius = 2 

Solidity =904.7808 cubic feet. 

2. The diameter of a sphere is 7957.8: what is its solidity 7 

Ans. 263863122758.4778. 

3. The diameter of a sphere is 24 yards : what is its solid 
contents ? Ans. 7238.2464 cubic yds. 

4. The diameter of a sphere is 8 : what is its solidity? 

Ans. 268.0832. 

a The diameter of a sphere is 16 : what is its solidity ? 

Ans. 2144.6656 

RULE II. 

Cube the diameter and multiply the number thus found, by t/u 
decimal .5236, and the product will be the solidity. 



EXAMPLES. 

f . What is the solidity of a sphere whose diameter is 20 ? 

Ans. 4188.8. 

2 What is the solidity of a sphere whose diameter is 6? 

Ans. 113.0976. 
3. What is the solidity of a sphere whose diameter is JO" 



22* 



Ans 523.6 



258 



APPLICATIONS 



Mensuration of t 



Round 13 o (1 i o s . 



PROBLEM X. 

To find the solidity of a spherical segment with one base.. 

RULE. 

I . T. three times the square of the radius of the base, add the 
square of the height. 

II. Multiply this sum by the height, and the product by the 
decimal .5236, the result will be the solidity of the segment. 

EXAMPLES. 

[ . What is the solidity of the seg- 
ment ABD, the height BE being 4 
feet, and the diameter AD of the 
base being 14 feet ? 

First, 

163: 



7 2 x-3 + 4 2 = 147+16 




Then, 163 x 4 x. 5236 = 341.3872 solid feet, which is the 
solidity of the segment. 

2. What is the solidity of the segment of a sphere whose 
neight is 4, and the radius of its base 8 ? Ans. 435.6352. 

3. What is the solidity of a spherical segment, the diam- 
eter of its base being 17.23368, and its height 4.5 ? 

Ans. 572.5566. 

4. What is the solidity of a spherical segment, the diam- 
eter of the sphere being 8, and the height of the segment 2 
feet ? Ans. 41.888 cubic Jt. 

5 What is the solidity of a segment, when the diame'er 
of the sphere is 20, and the altitude of the segment 9 feet ? 

Ans. 1781.2872 cubic ft 



OF GEOMETRY 



259 



Mensuration of the Spheioid 




OF THE SPHEROID. 

A spheroid is a solid described by the revolution of an 
ellipse about either of its axes. 

If an ellipse A CBD, be re- 
volved about the transverse or 
onger axis AB, the solid de- 
scribed is called a prolate 
spheroid : and if it be revolved 
about the shorter axis CD, the solid described is called an 
oblate spheroid. 

The earth is an oblate spheroid, the axis about which il 
revolves being about 34 miles shorter than the diameter per- 
pendicular to it. 

PROBLEM XI. 

To find the solidity of an ellipsoul 

RULE. 

Multiply the fixed axis by the square of the revolving axu,, 
and the product by the decimal .5236, the result will be the re- 
quired solidity. 

EXAMPLES. 

1. Tn the prolate spheroid 
ACBD, the transverse axis 
AB — 90, and the revolving 
axis CD = 70 feet: what is 
die solidity ? D 

Here, AB- 90 feet: CD* = 70 2 = 4900 : hence 
ABx CD 2 X. 5236 = 90 x 4900 x. 5236 = 230907.6 cubic feet., 
which is the solidity. 




2G0 



A PPLICATIONS 



Mensuration of Cylindrical Kings 



2. What is the solidity of a prolate spheriod, whoso fixod 
axis is 100, and revolving axis 6 feet ? Ans. 1 88 1.96. 

3. What is the solidity of an oblate spheroid, whose fix*>d 
axis is 60, and revolving axis 100 ? Ans. 314160 

4. What is the solidity of a prolate spheroid, whose d*e& 
are 40 and 50? Ans. 41888. 

5. What is the solidity of an oblate spheroid, whose axes 
are 20 and 10? Ans. 2094.4. 

6. What is the solidity of a prolate spheroid, whose axes 
are 55 and 33 ? Ans. 31361.022. 

7. What is the solidity of an oblate spheroid, whose axes 
are 85 and 75 ? Ans. 

OF CYLINDRICAL RINGS 

A cylindrical ring is formed by 
bending a cylinder until the two 
ends meet each other. Thus, if a 
cylinder be bent round until the axis 
takes the position mow, a solid will 
be formed, which is called a cylin- 
drical ring. 

The line AB is called the outer, and cd the inner diameter. 




PROBLEM XII. 

To rind the convex surface of a cylindrical ring. 

RULE. 

I. To the thickness of the ring add the inner diameter. 

II. Multiply this sum by the thickness, and the p'oduel 
9.8696, the result will be the area. 



OF GEO IYI E T R Y 



26 



Mensuration of Cylindrical Rings 




EXAMPLES. 

1 . The thickness A c, of a cylindri- 
cal ring is 3 inches, and the inner 
diameter cd, is 1 2 inches : what is 
(lie convex surface ? 

Ac+cd — 3 (-12 = 15: 
15x3x9.8696 = 444.132 square 
inches = the surface. 

2. The thickness of a c* lindrical ring is 4 inches, and the 
inner diameter 18 inches : what is the convex surface ? 

Ans. 868.52 sq. in. 

3. The thickness of a cylindrical ring is 2 inches, and the 
innnr diameter 18 inches ■ what is the convex surface ? 

Arts. 391.784 sq. tn. 

PROBLEM XIII. 

To find the solidity of a cylindrical ring. 

RULE. 

I To the thickness of a ring add the inner diameter 

II. Multiply this sum by the square of half the thickness, and 
[he product by 9.8696, the result will be the required solidity. 

EXAMPLES. 

i . What is the solidity of an anchor ring, whose inner di- 
ameter is 8 inches, and thickness in metal 3 inches ? 
84-3 = 11: then, 11 x (|) 2 X 9.8696=244.2?2G ; which ex- 
presses the solidity in cubic inches. 

2. The inner diameter of a cylindrical ring is 18 inches, 
and the thickness 4 inches : what is the solidity of the ring ? 

Ans. 868.5248 cubic inches 



262 APPLICATIONS 



Mensuration of Cylindrical Rings. 

3. Required the solidity of a cylindrical riiig whose thick* 
ness is 2 inches, and inner diameter 12 inches ? 

Ans. 138.1744 cubic in 

4. What is the solidity of a cylindrical ring, whose thick- 
ness is 4 inches, and inner diameter 16 inches? 

Ans. 789.568 cubic in. 

5. What is the solidity of a cylindrical ring, whose thick- 
ness is 8 inches, and inner diameter 20 inches ? 

Ans. 

6. What is the solidity )f a cylindrical ring whose thick 
ness is 5 inches, and inner diameter 18 inches ? 

Am\ 



A TABLE 



LOGARITHMS OF NUMBERS 

From 1 to 10,000 



H. 


Lo g- j 


N. 


Log. 


N. 

1 


Log. 


N. 
76 


Log. 


i 


o-oooooo 


26 


I-4U973 


5i 


1 -707570 


1 -880814 


2 


o-3oio3o 


3 


i-43i364 


52 


i-7i6oo3 


77 


1-886491 


3 


o-477»2i j 


1. 4471 58 


53 


1-724276 


78 


1-892085 


4 


0-602060 


29 


i-4'«2398 


54 


1-732394 


79 


1-897627 


5 


0-698970 


3o 


1-477121 


55 


1 -i4o363 


80 


1 -903090 


6 


0-778151 


3i 


i-49i362 


56 


1-748188 


81 


1 -908485 


I 


0-843098 


32 


1 -5o5i5o 


57 


I-755875 


82 


i- 9 .38i4 


0-903090 


33 


1 -5i85i4 


58 


1-763428 


83 


1 -919078 


9 


0-954243 


34 


1 -53 1 479 
1 • 544068 


5 9 


1 • 770852 


84 


1-924279 


10 


1 • 000000 


35 


60 


1 - 7781 5i 


85 


1 -929410 


ii 


1 041393 


36 


1 -5563o3 


61 


i-78533o 


86 


1.934498 


12 


l -079181 


11 


1-568202 


62 


1 -792392 


87 


1-939519 


i3 


1 -1 13943 


1-579784 


63 


1 -799341 


88 


1-944483 ' 


14 


1-146128 


39 


1 -591065 


64 


1 -806181 


89 


1.949.390 


i5 


1 • 1 7609 1 


40 


1 -602060 


65 


1 -812913 


90 


1 -954243 


16 


1 -204120 


41 


1-612784 


66 


1 -819544 


9 1 


1 -959041 


17 


1 • 23o44o 
1-250273 


42 


1 -623249 


67 


1-826075 


92 


1.963788 


18 


43 


1-633468 


68 


i-8325o9 


9 3 


1.968483 


'9 


1-278734 


44 


1 .643453 


69 


1-838849 


94 


1-973128 


20 


1 -3oio3o 


45 


1 -6532i3 


70 


1-845098 


95 


1.977724 


21 


1 -322219 


46 


1-662758 


71 


1 -85i258 


96 


1 .982271 


22 


1-342423 


47 


1 -672098 


72 


1-857333 


97 


1-986772 


23 


1-361728 


43 


1-681241 


73 


1-863323 


98 


1-991226 


24 


1 -380211 


49 


1 -690196 


74 


1-869232 


99 


1 .995635 


2D 


1-397940 


5o 


1 -698970 


75 


1-875061 


100 


2 • 000000 



Remark. — In the following table, in the nine right- 
hand columns of each page, where the first or lead- 
ing figures change from 9's to O's, points or dots are 
introduced instead of the O's, to catch the eye, and to 
indicate that from thence the two figures of the Log- 
arithm to be taken from the second column, stand in 
the next line below. 



z 


A T 


i.BLK 


OF 


L O (4 A KITH MS F R(J M 1 


TO 


10,00a 




N. 

100 


1 


I 


2J3l/ 4 |5j6l7|8 


9 


pq 

432 


000000 


0434 0868 i3oi! 1734 1 2166 2598 3029 3461 


38gi 


101 


432 1 


475ii 5i8ij 56og 6o38 6466 6894 7J2 1 ; 7748 


8.74 


428 


103 


8600 


9026J o45i; 9876. 8 3oo '724 1147 ^70 igg3 


*4i5 


424 


io3 


012837 


3259 368o 4100, 4521 1 4940 536o 5779 


6197 


6616 


419 


104 


7o32 


745i 


7868, 8284 


8700 01 16 g532, 9947 


•36i 


•775 


416 


io5 


02 1 1 89 


i6o3 


2016: 2428 


2841 


3252 


3f64 4075 


4486 


4896 


413 


106 


53o6 


5713 


6i25 6533 


6o42 


735o 


7-57; 8164 


8571 


8978 


408 


tci 


o384 
o33424 


9780' e ig5 # 6oo 
38s il 4227 4628 


1004 


1408 


l8l2 ; 2216 


2619 


3021 


404 


ro8 


5029 


543o 


583o: 623o 


662g 


7028 


4<>o 


i '°9 


7426 


7825! 8223, 8620 


9017 


0414 
3362 


981 1 *207 
3 7 55i 4148 


•602 


•998 


3 9 6 


:j o 


a4i3g3 


1787 2182 2576 


68 > 8? 


454o 


49-32 


3 9 3 
38 9 


in 


5323 


5714 


6io5 6495 
Q093 «38o 


7275 


7664 ! 8o53; 8442 


883o 


m 


0218 

05J078 


9606 
3463 


•766 


n53 i538 1924J 2309 


2694 


386 


n3 


3846 423o 


46i3 


4gg6| 5378 5760 6142 


6524 


382 


U4 


6905 


7286 


7666' 8046 


8426 


88o5! 9185 c563 
2582! 2g58 3333 


9942 


•320 


379 


i "J 


060698 


1075 


i452| 1829 


2206 


3709 
7443 


4o83 


376 


Ii6 


4438 


4832 


52o6' 558o 


5 9 53 


6326 


66gg' 7071 


78i5 


3 7 2 


"I 

ll3 


8180 


8557 


8928, 9298 


9668 
3352 


••38 


•4071 '776 


1 1 45 


i5i4 


36g 


071882 


325o 


2617 2985 


3 7 i8 


4085! 445 1 


4816 


5i82 


366 


! "j 


5547 


5912 
o543 

3 1 44 


6276; 6640 


7004 


7368 


773 1 8og4 


8457 


88 1 g 


363 


j 12 3 


079 181 


9904 1 *266 
3oo3 386i 


•626 


•987 


i347 1707 


2067 


2426 


36o 


121 


082780 


4219 


4576 


4g34 52gi 


5647 


6004 


35 7 


122 


636o 


6716 


707ij 7426 


7781 


8 1 36 8490 8845 


gig8 


o552 
J071 


355 


123 


9 9 o5 
093422 


•258 


•611 « 9 63 


i3i5 


1667 2018 2370 


2721 


35 1 


124 


3772 


4i22 4471 


4820 


5i6g! 55i8| 5866 


62i5 


6562 


34g 


125 


6910 
10037 1 


-2 5 7 


7604! 795i 


8298 


8644 8ggo 


g335 


0681 
3119 


••26 


346 


126 


0715 


ioSg 1403 


H47 


2ogi 


2434 


2777 


3462 


343 


! 127 

128 


38o4 


4146 


4487J 4828 


5169 
8565 


55io 


585i 


6191 


653 1 


6871 


340 


7210 


7549 


7888j 8227 


8go3 


9241 


9 5 79 


0916 
3275 


•253 


338 


129 


1 10590 


0926 


1263 


1 5 9 9 


1934 


2270 


26o5 


2g4o 


36og 


335 


i3o 


1 13943 


4277 


4611 


4944 


52 7 8 


56u 


5g43 


6276 
g586 


6608 


6940 


333 


1 3 1 


7271 


7603 


7934 


8265 


85 9 5 

1888 


8g26 


9256 


ogi5 
3 1 98 


•245 


33o 


l32 


120574 


0903 


I 23 I 


i56o 


2216 


2544 


2871 


3525 


328 


1 ^ 3 


3852 


4178 


45o4 


483o 


5i56 


548i 


58o6 


6i3i 


6456 


6781 


325 


1 i34 


7io5 


7429 


77 53 


8076 


83 99 


8722 


9045 


9368 


9690 


••12 


323 


i35 


i3o334 


o655 


0977 


1298 


i6iq 


l 9 3g 
5i33 


2260 


258o 


2900 


3219 
640J 


321 


1 36 


353q 


3858 


4H" 7 


4496 


4814 


545 1 


5769 


6086 


3i8 


'M 


6721 


7037 


7354 


7671 


7987 


83o3 


8618 


8 9 34 


924g 


9564 


3i5 


9879 


•t 9 4 


•5o8 


•822 


n36 


i45o 


1763 

4885 


2076 


238g 


2702 


3i4 


139 


u3oid 


332 7 


363 ? 


395i 


4263 


4574 


5ig6 


55o7 


58i8 


3i 1 


140 


146128 


6438 


6748 
9 835 


7o58 


736 7 


7676 


79 85 


82 9 4 


86o3 


891 1 


3og 


1 141 


9219 
152288 


9 5 27 


•142 


•449 


• 7 56 


io63i 1370 


1676 


1982 


3o 7 


142 


25g4 


2900 


32o5 


35io 


38r5 


4120 4424 


4728 


5o32 


3o5 


143 


5336 


5640 


5943 


6246 


6549 


6852 


7154 7457 


77 5 9 


8061 


3o3 


144 


8362 


8664 


8 9 65 


9266 


9 56 7 


g868 


•168 


•46g 


•769 


1068 


3oi 


u5 


i6i368 


1667 


1967 


2266 


2564 


2863 


3i6i 


346o 


3 7 58 


4o55 


299 


146 


4353 


465o 


4947 


5244 


554i 


5838 


6i34 


i.43o 


6726 


7022 


297 


147 


7317 


76i3 


7908 
0848 


8203 


I4o4 


8792 


9086 


g38o 


9 6 74 


9968 


293 


148 


170262 


o555 


1141 


1726 


2oig 23n 


26o3 


2895 


293 


149 


3i86 


3478 


3769 


4060 


435i 


4641 


4932! 5222 

7825 8n3 


55i2 


58o2 


291 


r5o 


1 7609 1 


638i 


6670 6959 


7248 


7536 


8401 


8689 
i558 


28g 


i5i 


8?77 


9264 


9^52 9839 


•126 


•4i3 


•6gg # g85 


1272 


287 


1 52 


l8l844: 2I2Q 


241 5, 2700 


2985 
5825 


3270 


3555 383g 


4i23 


4407 


285 


1 53 


4691 


4975 1 5259] 5542 


6108 


6391 6674 


6g56 


723g 


283 


1 54 


7521 


7803 


8084 8366 


8647 


8g28 


92og| g4go 


977 1 


•05, 


281 


1 55 


190332 


06 [2 


0892 1171 


I45i 


1730 


2010J 228g 


2567 


2846 


$ 


1 56 


3i25 


34o3 


368 11 395g 4237 


45i4 


4792I 5069 
7 556; 7832 


5346 


5623 


1 57 


58 99 


6176 


6453 


6729 7oo5j 7281 


8107 


8382 


276 


1 58 


86d 7 


8 9 32 


9206 


9481 1 Q755J »»2g 


•3o3; »577 


•85o 


1 1 24 


274 


159 

IN. 


201397 


1670 1943 


2216 2488 2761) 3o33 33o5 


35 77 


3848 


272 


i 1 1 2 


. 3 | 4 j 5 6 | 7 


8 9 I 


D. 





A TABLE 


OF 


LOGARITHMS FROM ] 


[ TO 


10,000. 


3 


NT. 
1 60 





1. 


2 


3 | 4 i 5 


6 


1 1 


8 ( 9 


3." 


204120 


4391 


4663 


4934 52o4 5475 


5746 


6016 


6286, 6556 


27 » 


161 


6826 


7096 


7365 


7634; 79°4 : 8173 


8441 


8710 


8979' 9247 


269 


162 


95i5 


9783 


••5 1 


•3ig ! «586 *853 


1 121 


1388 


1 654 


1921 


206 


1 63 


212188 


2454 


2720 


2986 3252 35i8 


3783 


4049 


43i4 


4579 


164 


4844 


5109 


53 7 3 


5638 5go2 6166 


643o 


6694 


6o5 7 
9 585 


7221 


264 


i65 


7484 


7747 


8010 


8273 8536 8798 


9060 


9323 


9846 


262 


166 


220108 


0370 


o63i 


0892 


n53 1414 


1675 


1936 
4533 


2106 


2456 


261 


16a 


2716 


2976 


3236 


3496 


3755 4oi5 


4274 


4792 


5o5i 


259 


5309 


5568 


5826 


60S4 6342 


6600 


6858 


71 id 


7372 


763oj 258 


l6«; 


7887 


8144 


8/oo 


865 7 8913 


9170 


9426 


9682 


99 38 


•193 256 


170 


330449 


0704 


oq6o 


1 2 r 5 1470 


1724 


1979 


2 234 


2488 


2742 254 


171 


2996 


325o 


3304 


3757 401 1 


4264 


45i7 


4770 


5o23 


5276 253 


172 


5028 


5 7 8i 


6o33 


6285 6537 


6789 


7041 


7292 


7544 1 779 5 l 25s 


I 7 3 


8046 


8297 


8548 


8799 9049 9299 


955o 


9800 


••5o *3oot 55o 


174 


240349 


0799 


1048 


1297 i546 1795 


2044 


2293 


254i 


2790 249 
52661 248 


l 7 5 


3o38 


3286 


3534 


3782 4o3o 4277 


4525 


4772 


5019 


176 


55i3 


5759 


6006 


6252 6499 6745 


6991 


7237 


7482, 77281 246 


177 


7973 


8219 


8464 


8709 8 9 54 i 9198 


9443 


9687 


9932: •176I 245 


178 


250420 


0664 


0908 


u5i 


i3 9 5 i638 


1881 


2125 2368 2610 243 


l l 9 


2853 


3096 


3338 


358o 


3822 4064 


43o6 


4548 


4790' 5o3i 


242 


180 


255273 


55i4 


5755 


5996 


6237 


6477 


6718 


6958 


71981 7439 


241 


181 


7679 


7918 


8i58 


83 9 8 


8637 


8877 


9116 


g355 


9594I 9833 


23 9 


182 


260071 


o3io 


o548 


0787 


1025 


1263 


i5oi 


1739 


1976 2214 


238 


1 83 


245 1 


2688 


2925 


3i62 


3399 


3636 


38 7 3 


4109 


4346' 4582 


23 7 


18; 


4818 


5o54 


5290 


5525 


5761 


5 99 6 


0232 


6467 


6702 1 6937 


235 


1 85 


7172 


7406 


7641 


7875 


8110 


8344 


8578 


8812 


9046 9279 


234 


186 


9 5i3 


9746 


9980 


•2l3 


•446 


•679 


•912 


1 144 


1377, 1609 


233 


187 


271842 


2074 


23o6 


2538 


2770 


3ooi 


3233 


3464 


3696 3927 


232 


188 


4i58 


438 9 


4620 


485o 


5o8i 


53u 


5542 


5772 


6002 6232J 23o 


189 


6462 


6692 


6921 


71 5 1 


7 38o 


7609 


7 838 


8067 


8296! 8525; 229 
•578j •800! 228 I 


190 


278754 


8982 


9211 


943o 
171 5 


9667 


9895 


•l23 


•35i 


191 


2 8io33 


1261 


1488 


1942 


2169 


23o6 
4606 


2622 


2849J 3o75 


227 


192 


33oi 


3527 


3 7 53 


3979 


42o5 


443 1 


4882 


5107 5332 


226 


193 


5557 


5782 


6007 


6232 


6456 


6681 


6905 


7i3o 


7354 7578 


225 


194 


7802 


8026 


8249 


8473 


8696 


8920 


9143 


9 366 


95891 9812 


223 


195 


290035 


0257 


0480 


0702 


0925 


1 147 


i36 9 


1 5 9 i 


i8i3 2o34 


222 


196 


2256 


2478 


2699 


2920 


3i4i 


3363 


3584 


38o4 


402DI 4246 


221 


197 


4466 


4687 


4907 


5127 


5347 


5567 


5787 


6007 
8.98 


6226 6446 


220 


i 9 8 


6665 


6884 


7104 


7 323 


7542 


7761 


7979 


8416 8635 


2IQ 


199 


8853 


9071 


9289 


9 5o 7 


9725 


9943 


•161 


•378 


•5 9 5 •8i3! 218 


200 


3oio3o 


1247 


U64 


1681 


1898 


21 14 


233i 


20471 2"?64j 2980I 217 


201 


3196 


3412 


3628 


3844 


4o5g 


4275 


4491 


4706 4921 5i36| 216 


202 


535 1 


5566 


5 7 8i 


5996 62 1 1 


642J 


6639 


6854 7068! 7282 2t5 


203 


7496 


7710 


7924 


8i3 7 ! 835i 


8564 


8778 


8991 9204J 9417 2l3 


204 


963o 


9843 


••56 


•268 «48i 


•6g3 


•906 


1118 i33o i542 2i2 


2o5 


311754 


1966 


2177 


2389 2600 2812 


3o23 


3234 3445 36561 211 


206 


3867 


4078 


4289 


4499 4710 4920 


5i3o 


5340 555i 


576c. 210 


207 


5970 


6180 


6390 
8481 


6599 6809 7018 
8689 8898 9106 


7227 


74361 7646 


7854! 200 
9938 208 


208 


8o63 


8272 


93 1 4 


9522 9730 


209 


320146 


o354 


o562 


0769 0977, 1184 


i3qf 


1 5 9 8 


i8o5 


2012 


207 


210 


322219 


2426 


2633 


2839 3o46 3252 


3458 


3665 


38 7 i 


4077 


2o6 


211 


4282 


4488 


4694 


4899 5io5 53io 


55i6 


5721 


5926 


6i3i 


205 


2!2 


6336 


654i 


6745 


6950 7 1 55 7359 


7 563 


7767 


7972 


8176 


204 


213 


838o 


8583 


878- 


899 1' 9194 s 9398 


9601 


9805 
i832 


•••8 


•21 li 203 


214 


33o4i4 


0017 


0819 


I022i 1225 1427 


i63o 


2o34 


2236; 202 


2l5 


2438 


2640 


2842 


3o44J 3246 ; 3447 


364q 


385o! 4o5i 


4253 202 


216 


4454 


4655 


4856 


5o57 5257 


5458 


-5658 


585 9 ! 6o5 9 


6260 201 


217 


6460 


6660 


6860 


7060 7260 


7459 


7 65 9 


7 85Sj 8o58 


8257; 200 1 


218 


8456 


8656 


8855 


9054 1 9203 945 1 


9600 


9849 ••47! «246! 199 


N. 


340444 


C642 


0S41 


1039 1237 1435 


[632 


i83o' 2028! 2225 


_L?t 





I 


2 


3 i 4 1 5 


6 


n 1 8 I 9 


D 



4 


A TABLE 


OP 


LOGARITHMS FROM 1 


TO 


10,000. 




N. 


| I | 2 


3 


4 5 | 6 


■> 


8 


9 




220 


342423| 2620 1 2817 3oi4 


32i 2( 3409' 36o6 


38o2 


3999 


4iq6 


221 


43o2 4589! 4785 
6353 i 6549' 6744 


4981 


5178 5374! 5570 


5766 


5962; 6137 


222 


6939 


7 135) 733o 7525i 7720 


7915, 8110 
9860 ••54 


195 


223 


83o5; 85oo 8694 


8889 


9083; 9278 9472 1 9666 


194 1 


224 


350248 


0442 


o636 


0829 


1023 12161 1410 i6o3 


1796 


1989 


1-3 


225 


2i83 


2375 


2568 


2761 


2954I 3i47 3339 
4876 5o68i 526o 


3532 


3724 


3916 


i 9 3 


226 


4108 


43oi 


4493 


4685 


5452 


5643 


5834 


192 


227 


6026 


6217 


6408 


6099 


6790' 6981' 7172 7363 


7554 


7744 


19? 


228 


7 o35 


8125 


83i6 


85o6 


8696; 8886 9076J 9266 


9456 


9646 


189 
188 


22Q 


9«35 ®*25 


•2l5 


•404 


•593: •7S3! ®972| 1 161 
2482 2671' 285g 3o48 


i35o 


i539 


23o 


361728 


1917 

3 800 


2105 


2294 


3236 


3424 


23l 


36i2 


3988 

5862 


4176 


4363 455 1 j 4739 4926 


5n3 53oi 


188 


232 


5488 


56 7 5 


6049 


6236 6423; 66jo' 6796 


6983, 7169 


187 


233 


7356 


7542 


7729 


79i5i 81 01 i 8287 


8473 865o 
•328, »5i3 


8845. go3o 


186 


234 


9216 


9401 


93871 9772 


99 58 


•i43 


•698 


•883 


1 85 


235 


371068 


1253 


1437J 1622 


1806 


1991 
383 1 


2175 


236o 


2544 


2/28 


184 


236 


2912 


3096 


328ol 3464 


3647 


4oi5 


4198 


4382 


4565 


184 


23 7 


4748 


4932 


5:i5| 5298 


5481 


5664 


5846 


6029 


6212 


63 9 4 


1 83 


238 


65 77 


6759 


6942! 7124 
8761 8943 


7306 


7488 


7670 
9487 


7852 


8o34 


8216 


182 


239 


83 9 8 


858o 


9124 


93o6 


9668 


9849 


••3o 


181 


240 


38o2 11 


0392 


0573 0754 


0934 


1 1 1 5 


1296 


1476 


i656 


i83 7 


181 


241 


2017 


3996 

5 7 85 


2377 2557 


2 7 3 7 


2917 


3o 97 


3277 


3456 


3636 


180 


242 


3Si5 


4174! 4353 


4533 


4712 


4891 


5070 


5249 


5428 


179 


243 


56o6 


5964 6142 


632i 


6499 


6677 


6856 


7o34 


7212 


'78 


244 


7390 


7568 


7746 7923 


8101 


8279 


8456 


8634 


8811 


8989 


^8 


245 


9166 


9343 


9320 9698 


9875 


••5i 


•228 


•4o5 


•382 


•739 


'77 


246 


390935 


1 112 


1288 


1464 


1 641 


1817 


1993 


2169 


2345 


2521 


176 


247 


2697 
4452 


2873 


3048 


3224 


34oo 


35 7 5 


375i 
55oi 


3926 


4101 


4277 


176 


248 


4627 


4802 


4977 


5i52 


5326 


5676 


585o 


6o25 


i 7 5 


249 


6199 


6374 


6548 


6722 


6896 
8634 


7071 
8808 


7245 


74i9 


7 5 9 2 


7766 


'74 


230 


397940 


8114 


8287 


8461 


8981 


9i54 


9328 


95oi 


i 7 3 


231 


9674 


9847 


••20 


•192 


•365 


•538 


•711 


•883 


io56 


1228 


i 7 3 


252 


401401 


1573 


•745| 1017 


2089 


2261 


2433 


26o5 


2777 


2949 


172 


253 


3l2I 


3292 


3464 


3635 


3807 


3 97 8 


4149 4320 


4492 


4663 


171 


254 


4834 


5oo5 


5176 


5346 


55i 7 


5688 


5858 


6029 


6199 


6370 


171 


255 


654o 


6710 


6881 


7o5i 


7221 


73 9 i 


7 56i 


7 7 3i 


7901 


8070 


170 


256 


8240 


8410 


8579 


8749 


8918 


9087 


9 25 7 


9426 


95 9 5 


9764 


169 


25t 


9933 


•102 


•271 


e 44o 


•600 

2 29 6 


•777 


•946 


1114 


1283 


i45i 


3 


258 


411620 


1783 


1956 


2124 


2461 


2629 


2796 


2964 


3i32 


239 


33oo 


3467 


3635 


38o3 


3970 


4i3 7 


43o5 


4472 


4639 
63o8 


4806 


167 


260 


4U973 


5i4o 


53o7 
6 973 


5474 


564i 


58o8 


5974 


6141 


6474 
8i35 


167 


261 


6641 


6807 


7i3o 


73o6 7472 
8964! 9129 


7 638 


7804 


7970 


166 


262 


83oi 


8467 


8633 


8798 
•45 1 


9 2 9 5 


9460! 9625 


9791 


i65 


263 


9956 *i 21 


©286 


•616 »78i 


•945 


mo 1275 


1439 


1 65 


264 


421604 1788 


1933 


2 °97 

3737 


2261 


2426 


2390 


2754 2918 
4392! 4555 


3o82 


164 


265 


3246 


3410 


3574 


3901 


4o65 


4228 


4718 


164 


?66 


4882 


5o45 


5208 


53 7 i 


5334 


5697 


586o 


6o23* 6186 


6349 

7973 


1 63 


2O7 


65 1 1 


6674 


6836 


6999 


7161 


7324 


7 486 


7648 7811 


162 


268 


8i35 


8297 


8459 

•» 7 5 


8621 


8783 


8944 


9106 


92681 9429 9591 


162 


269 


7752 


9914 


•236 


•398| e 559 


•720 


•88l I042| 1203 


161 


270 


43 ,364 


i525 


i685 


1846 


2007! 2167 


2328 


2488 2649 2809 


161 


271 


2969 
4669 
6i63 


3i3o 


3290 345oj 36io ! 3770 


3g3o 


4090 4249: 4409 


160 


272 


4729 


4888 


5o48 5207; 5367 


5326 


5685 5844! 6004 


1 5 9 


2 7 3 


6322 


6481 


6640 6798! 6957 


7116 


72-5 7433J 7592 


\U 


274 


77 5i 


7909 


8067 8226! 8384 8542 


8701 


8859 1 9017 9175 


275 


9 333 


9491 


9648 9806 9964 »I22 

1224! i38i 1338 1695 


•279 ®437j ^594! ^52 


1 58 


276 


440909 


1066 


1832 


2009 2166 2323 


1 5 7 


277 


2480 


2637 


2793J 2950 3io6 3263; 3419 
43371 45 1 3 4669 4825 1 4981 


3576 3732 3889 


! ?I 


2-8 


4045 4-01 


5 1 37 5293 1 5449 


i56 1 


?79_ 
N. 


3004 5760 1 591 5 6071 1 

~T~i~r| 7~i~| 


6226 6382| 6537 


6092 6848, 7003 


i55 


4 


5 6 


7 | 8 \ 9 | D. j 





A TABLE 


OF 


LOGARITHMS FROM 1 TO 


10,000. 


e 


N. 


1 J • 2 1 3 j 4 1 5 


1 6 

~8o~88 


1 * 


8 


9 


'""lv] 


280 


447158! t3i3 7468 7623, 777^1 7Q33 


8242 


8397 


8552 


1 55 


281 


870&J 8861 1 9015 


9170 9324 9478 9633 


9787 


994i 


•• 9 5 


1 54 


282 


450249! 04 ?3 


o557 


0711' o865| 1018 1172 


i3 2 6 


U79 


1 633 


i5 4 


283 


1 7861 1940 


2093 


2247 240c "J553 2706 


285 9 


3oi2, 3i65 


1 53 


284 


33i8j 3471 


3624 


3777, 393c, 4082 4235 


438 7 


454o, 4692 


1 53 


285 


4845 4997 ! 5i5o 


53o2 


: 5454 


1 56o6 575S 


5gio 


6062 


6214 


l52 


286 


6366 1 65i8' 6670 6821 


I 6 973 


| 7125, 7276 


7428 


7 5 79 


773i 


i5i 


287 


7882| 8o33| 8iS4 ! 8336' 8487! 8638 S789 


8940 


9091 


9242 


1 5 1 


288 


93921 g543 9694 9S45 9993 


1 •146, # 2g6 


•447 


•5 97 


1 # 748 


i5i 


289 


460898 1048 1 198 1 348 


1499 


j 1649 (799 


1948 


2098 


2 24& 


i5o 


29c 


462398 2548, 2697; 2847 


2997 


3i46 3296 


3445 


35 9 4 


3 7 44 


i5o 


291 


3893, 4042 


41 9 1 434o 


4490 463g 4788 


4936 


5o85 


5234 


149 


292 


5383 


5532 


568o 5829' 5977 


6126 6274 


6423 


6571 


6719 


\% 


293 


6868 


7016 


7164 73 1 2 ; 7460 


7608 7756 


7904 


8o52 


8200 


294 


8347 


8495 


8643 8790: 8938: go85, 9233 


9 38o 


9 52 7 


9675 


148 


29D 


9822 


9969 


•116 # 263! *4io 


•557, '704 


•85 1 


•998 


1 145 147 


296 


471292 


1438 


1 585, 1732; 1878 


2025 2171 


23i8 


2464 


2610! 146 


297 


2706 


2903 


3049 3i95 


334i 


3487, 3633 3779 
4g44' 5090' 5235 


3g25 


4071! 146 


298 


4216 


4362 


45o8 


4653 


4799 


538i 


5526 146 


299 


5671 


58i6 


5962 


6107 


6232 


6397 6542. 6687 


6832 


6976! 145 


3oo 


4771 2 1 


7266 ! 741 1 


7555 


7700 


7844; 7989' 8i33 


8278 
97'9 


8422! 145 


3oi 


8566 


87 uj 8855 


8999 


9U3 


9287, 943 1 j 9575 


9 863 j 144 


302 


480007 


oi5n 02g4 0438 


o582 


0725, 0869 1012 


n56 


1299 144 


3o3 


1443 


1 586 


1729 1872 


2016 


2l59 2302 2445 


2588 


2 7 3l 


143 


3o4 


2874 


3oi6 


3159' 33o2 


3445 


3587| 3730 3872 


4oi5 


4157 


143 


3o5 


43 00 


4442 


4585 | 4727 


4869 


5oii| 5i53 5295 


5437 


5579 142 


3o6 


5721 


5863 


6oo5 


6147 


6289 


643o 6572 6714 


6855 


6997, 142 


307 


7i38 


7280 


7421 


7563 


7704 


7845j 7986 8127 


8269 


84.101 141 


3o8 


855i 


8692 


8833 


8974 


9»4 


9255 9396 9537 


9 6 77 


9818 


141 


309 


9958 
491362 


• # 99 


•23 9 


»38o 


a ;)2o 


•661 j •8oi| »94i 


1081 


1222 


140 


3io 


l502 


1642 


17S2 


1922 


2062: 2201 j 2J41 


2481 


262 1 


140 


3u 


2760 


2900 


3o4o 


3i 79 


3319 


3458 3597I 3737 


3876 


401 5 


1 3 9 


3l2 


4i55 


4294 


4433 


4572 


4711 


485o 4989' 5i28 


5267 


54o6 


1 3 9 


3i3 


5544 


5683 


5822 


5960 


6099 


62381 6376 65i5 


6653 


6791 


1 3 9 


3i4 


6930 


7068 


7206 


7344 

8724 


7483 


7621 7759 1 7897 


8o35 


8n3 


i33 


3i5 


83 1 1 


8448 


8586 


8862 


8999I 9137 9275 


9412 


955o 


1 38 


3i6 


9687 


9824 


9962 


••99 »236 


•374J •Si 1 1 '648 


•785 


•922 


.37 


3i7 


5oio59 


1 196 


i333 


1470 ! 1607 


1744' 1880' 2017 


2 1 54 


2291 


i3 7 


3i8 


2427 


2564 


2700 2837! 2973 
4o63 4199 4335 


3io 9 ! 3246! 3382 


35x8 


3655 


1 36 


319 


3 79 i 


3 9 2 7 


447 1 1 4607 4743 


4878 


5oi4 


1 36 


320 


5o5i5o 


5286 


542 1 ! 5557 '• 5693 


5828 5 9 64i 6099 


6234 


6370 


1 36 


321 


65o5 


6640 


6776 691 ij 7046 


7181 7316 7451 
853o| 8664 1 8799 


7586 


7721 


i35 


322 


7856 


799 1 


8126; 8260; 83g5 


8q34 


9068 


i35 


3j3 


9>o3 


9337 


9471 9606 9740 


9874: •••9 1 «i43 


•277 


•411 


i34 


324 


5io545 


0679 


081 3i 0947 


1081 


I2i5j 1349I 1482 


1616 


1750 


i34 


325 


1 883 


2017 


2i5i j 2284 


2418 


255i 2684J 2818 


2951 


3o84 


i33 


3^6 


32i8 335i 


3484 36i7 


375o 


3883 4oi6 ; 4149 


4282 


44i4 


1 33 


32 7 


4548 


4681 


48 1 3 4946 


5079 


521 1 ! 53441 5476 


5509 


5741 


133 


328 


58 7 4 


6006 


6139 6271 


64o3 


6535 6668; 6800 


6 9 3 2 


7064 


132 


^ 9 


c l 1 - 96 


7328 


7460 7592 
8777 | 8 909 


7724 


7855 


7987! 81 19 


825i 


8382 


132 


33o 


5i85i4 


8646 


9040 


9171 


93o3 9434 9566 


9697 


i3i 


33i 


t 9 8 ?2 


9959 ••goj # 22I 


•353 


•48 * 


•6i5| «745| '876 


1007 


i3i 


333 


52ii38 


1269 1 i4oo ; i53o 


1661 


1792 1922J 2o53 2i83 


23i4 


i3i 


333 


2444 


25751 2705! 2835 


2966 


3o 9 6l 32261 3356| 3486 


36i6 


i3o 


334 


3746 


3876| 4006! 4i36 


4266 


4396 


4526 4656 4780 


4qi5 


i3o 


335 


5o45 


5i74j 53o4 5434' 5563 


56g3 
6 9 85 


5822J 595i 6081 


O210, 129 


336 


633 9 


6469' 65q3 6727 6856 


7ii4j 7 2 43 7372 


7001 129 


337 


763o 


7739 7888! 8016 1 8i45 


8274 


8402 853 1 8660 


8788' 12Q 


338 


8917 


9045, 9174 9302 943o 


9 55 9 


9687 j g8i5 9943 


••72; J 28 


339 


530200 o328; 0456; o584J 0712 


0840J O968: IO96 1 T223 


i35i| 128 


N. 


1 1 | 2 ; 3 4 


5 j 6 


7 1 3 . 


9 | D. 



6 


A TALSLK 


OF LOGARITHMS FROM 1 


TO 


L 0,000. 




N. 





1 | 2 


3 


4 


5 


6 1 7 


8 9 


D. 


34o 


53U79 


1607 


1734 


1862 


1990 


2117 


2245 ! 2372 


2D00 2627 


128 


34i 


2754 


2882 


3009 


3i36 


3264 


3391 


35 1 8 3645 


3772 38 99 


127 


342 


4026 


4i53 


4280 


4407 


4534 


4661 


4787 4914 


5o4i 5167 


127 


343 


5294 


5421 5547 


56 7 4 


58oo 


5927 


6o53| 6180 


63o6! 6432 


126 


344 


6558 


6685 681 1 


6 9 3 7 


7063 

8322 


11% 


73i5 
8574 


7441 


756 7 


7693 

8 9 5i 


126 


345 


7819 


7945 8071 


8197 


8699 


8825 


126 


346 


9076 


9202 


9327 


945a 


9 5 7 8 


9 7 o3 


9829 


99^4 


••79 


•204 


125 


347 

34« 


54o329 


0455 


o58o 


0705 


o83o 


0955 


1080 


I2o5 


i33o 


14^4 


125 


1579 


1704 


1829 


1953 


2078 


2205 


2327 


2452 


25 7 6 


271-11 


125 


549 


282! 


2950 


3074 


3i 99 


3323 


3447 


3571 


36 9 6 
4936 


3820 


3c>44 


124 


3jo 


544068 


4192 


43i6 


4440 


4564 


4688 


4812 


5o6o 


5i83 


124 


35i 


53o 7 


543 1 


5555 


5678 


58o2 


5925 


6049 


6172 


6296 


6419 


124 


35; 


6543 


6066 


6789 


6913 


7036 


7i5 9 


7282 


74o5 


7 52 ? 
8738 


7652 


123 


3; 53 


7775 


7898 


8021 


8144 


8267 


838 9 


85i2 


8635 


8881 


123 


354 


9003 


9126 


9249 
0473 


93 7 i 


9494 


9616 


97 3 9 


9861 


9984 


•106 


123 


355 


550228 


o35i 


0595 


0717 


0840 


0962 


1084 


1206 


i328 


122 


356 


i45o 


1572 


1694 


1816 


1938 


2060 


2181 


23o3 


2425 


2547 


122 


357 


2668 


2790 


2911 


3o33 


3 1 55 


3276 


33 9 8 


35i9 


364o 


3762 


121 


358 


3883 


4004 


4126 


4247 


4368 


4489 


4610 


473i 


4852 


4973 
6182 


121 


35 9 


5094 


52i5 


5336 


5457 


55 7 8 


5699 


5820 


5 9 4o 


6061 


121 


36o 


5563o3 


6423 


6544 


6664 


6785 


6905 


7026 


7146 


7267 


738 7 


I20 


36 1 


7D07 


7627 


7748 
8948 


7868 


7988 


8108 


8228 


8349 


8469 


858 y 


120 


36a 


8709 


8829 


9068 


9188 


93o8 
•5o4 


9428 


9548 


9667 


97B7 


120 


363 


9907 


••26 


•146 


•265 


•385 


•624 


•743 


•863 


•982 


119 


364 


56 1 1 01 


1221 


1 34o 


U59 


1578 


1698 


1817 


1936 


2o55 


2174 


II9 


365 


2293 


2412 


253i 


265o 


2769 
3955 


2887 


3oo6 


3l2D 


3244 


3362 


119 


366 


348i 


36oo 


3718 


3837 


4074 


4192 


43u 


4429 


4548 


H9 


36 7 


4666 


4784 


4903 


5021 


5i39 


5257 


5376 


5494 


56i2 


5730 


Il8 


368 


5848 


5966 


6084 


6202 


6320 


6437 


6555 


66 7 3 
7849 


6791 


6909 


Il8 


369 


7026 
568202 


7144 


7262 


7379 


7497 
8671 


7614 
8788 


7732 


7967 


8084 


Il8 


370 


83i 9 


8436 


8554 


8 9 o5 


9023 


9140 


9257 


117 


3 7 i 


9374 


9491 


9608 


9725 


9842 


99 5 9 


••76 


•, 9 3 


•3o 9 


•426 


117 


3 7 2 


570543 


0660 


0776 


0893 


IOIO 


1 1 26 


1243 


i3^o 

2523 


1476 


i5o2 

2755 


>'7 


3 7 3 


1709 


i825 


1942 


2o58 


2174 


2291 


2407 
3568 


2639 


116 


374 


2872 


2988 


3 1 04 


322o 


3336 


3452 


3684 


38oo 


3 9 i5 


116 


3 7 5 


4o3i 


4i47 


4263 


4379 


44Q4 
565o 


4610 


4726 


4841 


4957 


5072 


116 


3 7 6 


5i88 


53o3 


5419 


5534 


5 7 65 


588o 


5996 


6111 


6226 


u5 


377 


634i 


6457 


65 7 2 


6687 


6802 


6917 
8066 


7032 


7147 


7262 


7377 


1 1 5 


378 


7492 


7607 


7722 


7836 
8 9 83 


79 5i 


8181 


82 9 5 


8410 


85?5 


1 1 5 


379 


8639 


8 7 54 


8868 


9097 


9212 


9 326 


9441 


9555 


9669 


114 


38o 


5 79 7«4 


9898 


••12 


•126 


•241 


•355 


•469 


•583 


•697 
1 836 


•811 


114 


38i 


580925 


1039 


u53 


1267 


i38i 


I4g5 
263 1 


1608 


1722 


1950 


114 


382 


2o63 


2177 


2291 


2404 


25i8 


2745 2858 


2972 


3o85 


114 


383 


3199 


33i2 


3426 


353 9 


3652 


3765 


38 79 


3 99 2 


4io5 


4218 


1 13 


384 


433 1 


4444 


4557 


4670 


4783 


4896 


5009 


5l22 


5235 


5348 


n3 


385 


5461 


5574 


5686 


5799 


5912 


6024 


6i3 7 


6250 


6362 


6475 


n3 


386 


6587 


6700 


6812 


6925 


7037 


7i49 


7262 
8384 


7374 


7486 


7599 


U2 


387 


7711 


7823 


7935 


8047 


8160 


8272 


8496 


8608 


8720 


112 


388 


8832 


8944 


9o56 


9167 


9279 


9391 


95o3 


96l5 


9726 


9838 


112 


38 9 


9950 


••61 


•i 7 3 


•284 


•3 9 6 


•5o 7 


•619 


•730 


•842 


• 9 53 


112 


390 


591065 


1176 


1287 


l3 99 


i5io 


1621 


1732 i843 


1955 


2066 


III 


3 9 i 


2177 


2288 


2399 


25io 


2621 


2 7 32 


2843 


2954 


3o64 


3i 7 5 


III 


3 9 2 


3286 


33 9 7 


35o8 


36i8 


3729 


3840 


3 9 5o 


4o6l 


4171 


4282 


III 


3 9 3 


4393 


45o3 


4614 


4724 


4834 


4945 


5o55 


5i65 


5a 7 6 


5386 


no 


? 9 i 


5496 


56o6 


5717 


5827 


5 9 3 7 


6047 


6i5 7 


6267 


63 77 


6487 


no 


3 9 5 


65 97 
7695 
8791 


6707 


6817 


6927 


7037 


7U6 
8243 


7256 


7366 


7476 


7586 


no 


3 9 6 


7 8o5 


7914 8024 


8i34 


8353 


8462 


8572 


8681 


no 


1 l 9 l 


8900 


9009 91 19 


9228 


9337 


9446 


9556 


9 665 


9774 


109 


3 9 8 


9 883 


999 2 


•iOIJ *2I0 


•319 
1408 


•428 


•53 7 


•646 


•755 


•864 


109 


1 399 
* N. 


600973 


I082 I I9I ! I2Q9 


i5i7 


i625 


1734 


i843! 1901 


109 





1 ! > ! 3 


4 


5 


6 


7 


~8~yv 


D. 





A TkBLE 


OF 


LOGARITHMS FR 


OM ] 

6 


TO 


10,000. 


* 


N. 
400 





1 


~ T_ 


3 


2494 


2603 


7 


8 


JL 


D. 


602060 


2169 
3253 


2277 


2386 


2711 


2819 


2928 


3o36 


108 


401 


3i44 


336 1 


3469 


3577 
4658 


3686 


3794 


3902 


4010 


4118 


108 


402 


4226 


4334 


4442 


455o 


4766 


4874 


4982 


3089 


5197 


108 


4o3 


53o5 


54i3 


552i 


5628 


5 7 36| 5844 


5951 


6059 


6166 


6274 


108 


404 


638 1 


6489 


65 9 6 


6704 


681 1 


6919 


7026 


7 i33| 7241 


7348 
8419 


107 


403 


7455 


7362 


7669 
8740 


7777 
8847 


7884 


7991 


8098 


8 2 o5 


83i2 


07 


406 


8526 


8633 


8934 


9061 


9167 


9274 


9 38i 


9488 


107 


407 


9394 


9701 


9808 


9914 


••21 


•128 


•234 


•341 


•447 


•554 


107 


408 


610660 


0767 


0873 


0979 


1086 


1192 


1298 


i4o5 


1 5i 1 


1617 


106 


409 


1723 


1829 


1936 


2042 


2148 


2234 


236o 


2466 


2572 


2678 


106 


410 


612784 


2890 


2996 


3l02 


3207 


33i3 


3419 


3525 


363o 


3736 


106 


411 


3842 


3947 


4o33 


4159 


4264 


4370 


4475 


458 1 


4686 


4792 


106 


412 


4897 


5oo3 


5io8 


5 2 i3 


5319 


5424 


5329 


5634 


5740 


5845 


io5 


4i3 


5930 


6o55 


6160 


6265 


6370 


6476 


658i 


6686 


6790 


68 9 5 


io5 


4U 


7000 


7io5 


7210 
825-1 


73i5 


7420 


7325 


7629 


7734 


7 83 9 


7943 
8989 


io5 


41 5 


8048 


8i53 


8362 


8466 


85 7 i 


8676 


8780 


8884 


io5 


416 


9093 


9198 


9302 


94*06 


95i 1 


9 6i5 


9719 


9824 


9928 


••32 


104 


417 


620106 


0240 


o344 


0448 


o552 


o656 


0760 


0864 


0968 


1072 


104 


418 


1 176 


1280 


1 384 


1488 


1592 


i6o5 
2732 


'799 


igo3 


2007 


21 10 


104 


419 


2214 


23i8 


2421 


2525 


2628 


2833 


2g3o 
3973 


3o42 


3i46 


104 


420 


623249 


3353 


3456 


3559 


3663 


3766 


386 9 


4076 


4H9 


io3 


421 


4282 


4385 


4488 


4591 


4695 


479S 


4901 


5oo4 


5i07 


5210 


io3 


422 


53i2 


54i5 


55i8 


562i 


6724 


5827 


5929 


6o32 


6i35 


6238 


io3 


423 


634o 


6443 


6546 


6648 


6 7 5 1 


6853 


6956 


7038 


7161 


7263 


io3 


424 


7366 


7468 


7^71 


7673 7775 


7878 


7980 


8082 


8i85 


8287 


102 


425 


838 9 


8491 


85 9 3 


86 9 5 


«797 


8900 


9002 


9104 


9206 


9 3o8 


102 


426 


9410 


9312 


961 J 


9713 


98'7 
o835 


9919 


••21 


•123 


•224 


•326 


102 


427 


63o428 


o53o 


o63i 


0733 


0936 


io38 


1 139 
2i53 


1241 


i342 


102 


428 


1444 


1 545 


1647 


1748 


1849 


1961 


2052 


2235 


2356 


101 


429 


2457 


2559 


2660 


2761 


2862 


2963 


3o64 


3i65 


3266 


336 7 


101 


43o 


633468 


3369 


3670 


3771 


38 7 2 


3973 


4074 


4n5 
5i82 


4276 


43 7 6 


ioo 


43 1 


4477 
5484 


4378 


4679 
3685 


4779 
5 7 85 


4880 


4981 


5o8i 


5283 


5383 


100 


432 


5584 


5886 


5 9 86 


6087 


6187 


6287 


6388 


ioo 


433 


6488 


6588 


6688 


6789 


6889 


6989 


1089 
8090 
9088 


7.89 


7290 


73oo 
838 9 
9 38 7 


ICO 


434 
435 


7490 
8489 


858 9 


7690 
8689 


fiX 


7890 

8888 


7990 
8988 


8190 
9188 


8290 
9287 


99 
99 


436 


94S6 


9586 


9686 


9783 


9885 


99B4 


••84 


•i83 


•283 


•382 


99 


437 
438 


640481 


o58i 


0680 


0779 


0879 


0978 


1077 


"77 


1276 


i375 


99 


1474 


i573 


1672 


1771 


1871 


1970 


2069 
3o58 


2168 


2267 


2366 


99 


43 9 


2465 


2563 


2662 


2761 


2860 


2939 


3i56 


3255 


3354 


$ 


44o 


643453 


355i 


365o 


3 749 


3847 


3 9 46 


4044 


4i43 


4242 


434o 


44 1 


4439 


4537 


4636 


4734 


4832 


493i 


5029 


5 1 27 


5226 


5324 


98 


442 


5422 


5521 


5619 


5 7 i-7 
6698 


58i5 


5913 

6894 


601 1 


6i 10 


6208 


63o6 


98 


443 


6404 


65o2 


6600 


6796 


6992 


7089 


•7187 


7285 


98 


444 


7383 


7481 
8458 


2 5l 9 


7676 


7774 


7872 
8848 


7969 
8945 


8067 


8i65 


8262 


98 


445 


836o 


8553 


8653 


8 7 5o 


9043 


9140 


9237 


97. 


446 


9 335 


9432 


953o 


9627 


9724 


9821 


9919 
0890 


••16 


•1 13 


•210 


97 


447 
448 


65o3o8 


o4o5 


0502 


0599 


0696 


0793 


0987 


1084 


1181 


97 


1278 


1375 


1472 


1569 


1666 


1762 


1839 


1936 


2o53 


2i5o 


97 


449 


2246 


2343 


2440 


2536 


2633 


2730 


2826 


2923 


3019 


3i 16 


97 


45o 


6532i3 


3309 
42 7 3 


34o5 


35o2 


35 9 8 


36 9 5 


3791 

4754 


3888 


3 9 84 


4080 


96 


431 


4177 


436 9 


4465 


4562 


4658 


485o 


4946 


5o42 


96 


452 


5i38 


5 2 35 


533 1 


5427 


5523 


5619 


5 7 i5 


58io 


5906 
6864 


6002 


96 


453 


6098 


6194 


6290 


6386 6482 


65 77 


6673 


6769 
7725 


6960 


96 


454 


7036 


7i52 


7247 


7343! 7438 


7 534 


7629 
8584 


t: 


7916 
8870 


9 6 


455 


801 1 


8107 


8202 


8298I 83 9 3 
92301 9346 


8488 


8679 


9 5 


456 


8 9 65 


9060 


9i55 


9441 


9 536 


9 63i 


9726; 9821 


^ 


457 
458 


9016 
66oS65 


••11 


°io6 


•201 1 ^296 


•3oi 
1 33 9 


•486 


•58i 


•676 0771 


9 5 


0960 


1033 


iiSol 1245 


1434 


i52o 
2475 


1623 1718 


9 5 


459 


i8i3 


1907 


2002 


2096! 2191 


2286 


238o 


256 9 2663 


93 


N. 





I 


2 


3 | 4 


* 


6 


7 


8 j 9 


I>. 



a:^ 



tf 


A TABLE 


OF 


LOGARITHMS FROM 1 


TO 


10,000. 




N. 





1 


2 


3 | 4 | 5 j 6 | 7 J 8 | 9 


j-b7 


460 


662708 


2852 


2947 


3o4ii 3i35 323o| 3324; 34i8 


35I2 1 3607 94 


461 


3701 


3795 


3889 


39831 4078! 4172] 4266 436o 


4454j 4548 


94 


462 


4642 


4736 


483 


4924 ! *5oi8 
5862 5956 


5ii2 52o6! 5299 
6o5o ! 6i43 6237 


5393: 5487 


94 


463 


558 1 


5675 5769 


633 1 6424 


94 


464 


65i8 


6612 


6705 


6799 68 9 2 


6986 7079! 7 i 7 3 


7266! 7360 


94 


465 


7453 


7546 


7640 


77331 7826 
8665! 8759 


7920 8oi3l 8106 


8199! 8293 ! 93 


466 


8386 


8479 


8372 


8852 


8945 


9 o38 


9i3ij 9224I 93 


467 


9317 


9410 g5o3 


9596 


9689 


9782 


9875 


9967 

0895 


••60! »i53 93 


468 


670246 


o339 
1265 


043i 


o524 


0617 


07101 0802 


0988 


1080 


93 


469 


1173 


1358 


145 1 


1 543 


1636 


1728 


1821 


iqi3 


2oo5 


9 3 


470 


672098 


2190 


2283 


23 7 5 


2467 


256o 


2652 


2744 2836 


2929 


92 


47i 


3021 


3n3 


32o5 


3297 


3390 


3482 


3574 


3666 


37D8 


385o 92 


47^ 


3942 
4861 


4o34 


4126 


4218 


43 10 


4402 


4494 


4586 


4677 


4769 92 


4 7 3 


4953 

58 7 o 


5o45 


5i3 7 


5228 


5320 


5412 


55o3 


5595 


568 7 


92 


474 


6694 


5962 


Oo53 


6145 


6236 


6328 


6419 


65n 


6602 


92 


475 


6785 


6876 


6968 


7o5 9 


71D1 


7242 
8i54 


7 333 


7424 
8336 


75i6 


9' 


476 


760-7 


7698 


7789 


7881 


7972 
8882 


8o63 


8245 


8427 


91 


477 


85i8 


8609 


8700 


8791 


8 97 3 


9064 


oi55 


9246 


9 33 7 


9' 


478 


9428 


9 5l 9 


9610 


9700 


979 « 


9882 


9973 ««63 


•i54 


•245 


9i 


479 


68o336 


0426 


o5i7 


0607 


0698 


0780 


0879 


0970 


1060 


1 1 5 1 


9 1 


480 


681241 


i332 


1422 


i5i3 


i6o3 


1693 


1784 


1874 


1964 


2o55 


90 


481 


2145 


2235 


2326 


2416 


25o6 


2596 


2686 


2777 


2867 


2957 


90 


482 


3o47 


3i3 7 


3227 


33i 7 


3407 


3497 


3587 


36 7 7 


3767 


385 7 


90 


483 


3947 


4037 


4127 


4217 


43o7 


43q6 


4486 


4576 


4666 


4756 


90 


484 


4845 


4935 


5o25 


5ii4 


52o4 


5294 


5383 


5473 


5563 


565 2 


si 


485 


5742 


583 1 


5921 


6010 


6100 


6,69 


6279 


6368 


6458 


6547 


486 


6636 


6726 


68i5 


6904 


6994 

7886 


7o83 


7172 
8064 


7261 


735i 


744o 


i 9 


487 


7 52 9 


7618 


7707 
85 Q 8 


7796 


7975 
8865 


8i53 


8242 


833 1 


89 


488 


8420 


8509 


8687 


8776 


8 9 53 
0841 


9042 


9i3i 


9220 


89 


489 


9309 


9 3n8 


9486 


9575 


9664 


97 53 


99 3o 


••10 

090D 


•107 


S o 9 


490 


690 1 96 
1081 


0285 


o373 


0462 


o55o 


0639 


0728 


0816 


^993 


8 o2 


491 


1 170 


1258 


1 347 


1435 


i524 


1612 


1700 


1789 


1877 


88 


492 


196D 


2o53 


2142 


2230 


23i8 


2406 


24Q4 


2583 


2671 


2759 


88 


493 


1847 


2o35 
38i5 


3o23 


3i 1 1 


3 '99 


3287 


3375 


3463 


355i 


363 9 


88 


494 


3727 


3903 


3 99 i 
4868 


4078 


4166 


4254 


4342 


443o 


4517 


88 


495 


46o5 


4693 


4781 


4 9 56 


5o44 


5i3i 


5219 


5307 


53 9 4 


88 


496 


5482 


556q 


5657 


5744 


5832 


5 9 i 9 


6007 


6094 


6182 


6269 


£ 7 


497 


6356 


6444 


653 1 


6618 


6706 


6 79 3 


6880 


6968 


7o55 


7142 


? 7 


498 


7229 


73i7 
8188 


7404 


749i 


7 5 7 8 
8449 


7665 


77 52 


7 83 9 


7926 


8014 


S 7 


499 


8101 


8275 


8362 


8535 


8622 


8709 


8796 


8883 


^ 


. 5oo 


698970 


9057 


9144 


923i 


9317 


94o4 


9491 


9 5 7 8 


9664 


975 1 


87 


5oi 


9838 


9924 


••11 


••98 


•184 


•271 


•358 


•444 


•53 1 


•617 


s 


502 


700704 


0790 


0877 


0963 


io5o 


n36 


1222 


1 309 


i3 9 5 


1482 


5o3 


1 568 


1 654 


1741 


1827 


1913 


1999 


2086 


2172 


2258 


2344 


86 


5o4 


243 1 


2517 


26o3 


2689 


2 77 5 


2861 


2947 


3o33 


3i 19 


32o5 


86 


£o5 


32qi 
4i5i 


3377 


3463 


3549 


3635 


3721 


3807 


38 9 3 
475i 


3979 


4o65 


86 


5o6 


4236 


4322 


4408 


4494 


4579 


4665 


4837 


4922 


86 


507 


5oo8 


5094 


5179 
6o3d 


5265 


535o 


5436 


5522 


5607 


56 9 3 


5 77 8 


86 


5o8 


5864 


68o3 


6120 


6206 


6291 


63 7 6 


6462! 6547 


6632 


85 


509 


6718 


6888 


6974 7059 


7144 


T229 


7 3i5 


74oo 


7485 
8336 


85 


5io 


707570 


7655 


7740 
85 9 i 


7826 


791 1 


7996! 8081 


8166 


825i 


85 


5n 


8421 


85o6 


8676 


8761 


8846 8 9 3i 


9015 


9100 


9 i85 


85 


5l2 


9270 


9 355 


9440 


9524 


9609 


9694 9779 


9 863 


9948 


••33 


85 


5i3 


710117 


0202 


0287 


0371 


0456 


0040, 0625 


0710 


0794 


0879 


85 


5i4 


0963 


1048 Il32 


1217 


i3oi 


1 385 1470 


1 554i 1639 


1723 

2566 


84 


5i5 


1807 


1892 


1976 
2818 


2060 


2144 


2229 


23i3 


2397 2481 


84 


5i6 


2o5o 


2734 


2902 


2986 
3826 


3070 


3i54 


3238 


3323 


3407 


84 


5l I 
5i8 


349i 


35i5 


3659 


3742 

458i 


3910 


3994 

4833 


4078 


4162 


4246 


84 


433o 4414 


4497 


4665, 


4749 


4916 


5ooo 


0084 


84 


619 


5167 525i 


5335 


54i8 


55o2 


5586 


5669 


5 7 53 


5836 


5 9 2o 


84 
D. 


jr. 


J 1 


2 


3 


4 ! 5 


6 


7 


8 J 9 j 





A TABLE 


OF 


LOGARITHMS FROM 


L TO 


10,000. 


9 


N. 


1 1 . 1 2 1 3 j 4 


5 


6 j 7 j 8 | 9 

1 ' 


-xi 


520 


716003 6087; 6170) 6254; 6337 


6421 


65o4, 6588 667 il 6754 


S3 


521 


6838 6921; 7004] 7088] 7171 


7204 


7338 


7421 


75o4 7^87 


83 


522 


7671 


7754] 7837 792c 8oo3 


8086 


8169 


8253 


8336 8419 


83 


523 


85o2 


8585! 8668 8 7 5i 8834 


8917 


9000 1 9083! 9 1 651 9248 


83 


524 


933i 


94i4i 9497 j 958oi 9663 


9740J 9828 991 1 1 99941 ee 77 


83 


525 


720159 


0242, o325 04071 0490 


o573 ; o655i 0738! 0821! ogo3 


83 


626 


0986 


1068 


ii5i 


1233 i3i6 


i3 9 8 


148 1 


1 563. 1646 1728 


*2 


52 7 


1811 


1893 


1975 


2o58: 2140 


2222 


2«io5 


2387 2469; 2552 


i* 


528 


2634 


2716 


2798 


2881 2 9 63 
3702! 3784 


3o45 


3127 


3209 3291 3374 


82 


529 


3456 


3538 


3620 


3866 


3948 


4o3o 41 12, 4iq4 


82 


53o 


724276 


4358 


4440 


452 2 i 4604 


4685 


4767 


4849 493 1 


5oi3 


82 


53 1 


5095 


5176 


5258, 534o| 5422 


55o3 


5585 


56 il 5748 


583o 


82 


532 


5912 


5993 


6075 6i56 6238 


632o 


6401 


6483 


6564 


6646 


82 


533 


6727 


6809 
7623 


6890 697 2 1 7o53 


7i34 


7216 


7297 
8110 


7379 


746o 


81 


534 


7^41 


7704 


77801 7866 
8097 8678 


7948 


8029 


8191 


8273 


81 


5*5 


8354 


8435 


85i6 


8759 


8841 


8922 


9003 


9084 


81 


536 


9165 


9246 


9327 


9408 9489 


9 5 7 o 


965i 


97 32 


9 8i3 


9893 


81 


53 7 


9974 


••55 


. •i36 


•217 *298 


•3 7 8 
1 186 


•459 


•540 


•621 


•702 


81 


538 


730782 


o863 


0944 


1024' no5 


1266 


1 347 


1428 


i5o8 


81 


539 


1589 


1669 


1750 i83o| 191 1 


1991 


2072 


2l52 


2233 


23i3 


81 


54o 


732394 


2474 


2555 


2635 2715 


2796 


2876 


2 9 56 


3o37 


3117 


80 


541 


3i 97 


3278 


3358 


3438 35i8 


3598 


3679 


3759 


383 9 


3919 


80 


543 


3999 


4079 


4160 


4240] 4320 


4400 


4480 


456o 


4640 


4720 


80 


543 


4806 


4880 


4960 


5o4oi 5 1 20 


5200 


5279 
6078 


5359 


5439 


55i9 


80 


544 


6599 


56 79 


5 7 59 


5838 5 9 i8 


5998 


6157 


623 7 


63i 7 


80 


540 


6397 


6476 


6556 


6635! 6715 


6795 


6874 


6954 


7o34 


7 1 13 


80 


545 


7193 


7272 


7352 


743 i| 75 1 1 


7390 
8384 


7670 


7749 


7829 
8622 


7908 


79 


54? 


79^7 
8781 
9572 


8067 


8146 


8220 83o5 


8463 


8543 


8701 


79 


048 


8860 


8 9 39 


9018 9097 


9177 


9256 


9 335 


94i4 


9493 


79 


54 9 


965 1 


97 3i 


9S10J 9889 
0600! 0678 


9968 


••47 


•126 


•205 


•284 


79 


5do 


74o363 


0442 


0521 


0757 


o836 


0915 


0994 


1073 


79 


55i 


I 102 


1230 


i3o9 


1 388 1467 


1 546 


1624 


1703 


1782 


i860 


79 


552 


1939 


2018 


2096 2175 2254 


2332 


241 1 


2489 


2568 


2647 


31 


553 


2725 


2804 


2882 


2961 


3o39 
38 2 3 


3n8 


3 196 


3275 3353 


343 1 


554 


35io 


3588 


3667 


3745 


3902 


3980 


4o58 


4i36 


42i5 


7» 


555 


4293 


4371 


4449 


4528 


4606 


4684 


4762 


4840 


4919 


4997 


78 


556 


5075 


5i53 


5 2 3i 


53o9 


5387 


5465 


5543 


562i 


0699 


5777 


78 


55 7 


5855 


5 9 33 


60 1 1 


6089 


6167 


6245 


6323 


6401 


6479 


6556 


78 


558 


6634 


6712 


6790 


6868 


6945 


7023 


7101 


7H9 


7256 


7334 


78 


539 


74i2 


7489 
8266 


7567 


7645 


7722 


7800 


7878 


79 55 


8o33 


8110 


78 


56o 


748188 


8343 


8421 


8498 


85 7 6 


8653 


8731 


8808 


8885 


77 


56i 


8 9 63 


9040 


9118 


9 i 9 5 


9272 


935o 


9427 


95o4 


9 582 


9 65 9 


77 


562 


9736 


9814 


989 1 


9968 


••45 


•123 


•200 


•277 


•354 


•43 1 


77 


563 


75o5o8 


o586 


o663 


0740 


0817 


0894 


09-1 1 


1048 


1123 


1202 


77 


564 


1279 


1 3 56 


1433 


i5io 


i5S7 


1664 


1 74 1 


1818 


1890 


1972 


77 


565 


2048 


2125 


2202 


2279 


2356 


2433 


2509 


2 586 2663 


2740 


77 


566 


2816 


28 9 3 


297O 


3o47 


3i23 


3200 


3277 


3353 


3430 


35o6 


77 


D67 


3583 


366o 


3 7 36 


38i3 


388 9 


39661 4042 


4119 


4195 


4272 


77 


568 


4348 


4425 


45oi 


45 7 8 


4654 


4730! 4807 


4883 


4960 


5o36 


76 


56 9 


5ll2 


0189 


5 2 65 


5341 


5417 


5494 5570 


5646 


5722 


5799 


76 


570 


755875 


5951 


6027 


6io3 


6180 


6256 6332 


6408 


6484 


656o 


76 


i 11 


6636 


6712 


6788 


6864 


6940 


70i6| 7092 


7168 


7244 

8oo3 


7320 


76 


572 


7396 
8i5d 


7472 


7548 


7624 


7700 


■7775, 7801 


7927 


8079 


76 


573 


823o 83o6 


8382 


8458 


8533! 8609 


8685 


8761 


8836 


-6 


5n4 


8912 


8988 9063 


9 i3 9 


9214 


0290I 9366 


944i 


9517 


q5q2 


16 


575 


068 


9743 1 9819 


9894! 997° 


••45! «I2I 


•196 
0920 


•272 


•347 


75 


57& 


760422 


0498 j o573 


0649 0724 


0799! 0870 


1025 


1:01 


7 ? 


t\l 


1176 


I2DI 1326 


1402 1477 


IDD2I 1627 

23o3l 2378 


1702 


1778; i853 


75 


Ig28 


2oo3J 2078 


2i53 2228 


2453 


25291 2604 
32 7 8j 3353 


75 


i 579 


2679 


2754 2829 


2904! 2978 


3oo'3 3128 


32o3 


75 


L*. 





I 


2 | 3 


_^J 


5 1 

1 


6 


_7__ 


8 1 


9 


D. 



10 


A TABLE 


OF 


LOGARITHMS FROM 1 TO 10,000. 




- N : 





1 


2 

"35^8 


3 | 4 | 5 


6 1 7 1 8 I 9 


1). 


58o 


763428 


35o3 


3653! 3727 1 38o2 


3877 3952I 4027 4101 


75 


58i 


4176 


425i 


4326 


44oo 44761 455o 


4624 4699' 4774 4848 


75 


582 


4923 


4098 


5072 


5i47 522i| 5296! 5370 3445 552o| 5394 


75 


583 


566 9 


5743 


58i8 


58 9 2 i 5 9 66i 6041 
6636 6710! 6785 


6n5j 6190 6264 
6839' 6933 7007 


6338 


74 


584 


64l3 


6487 


6562 


7082 


74 


585 


7i56 


723o 


73o4 


7379 1 7433 7527 


7601 


7675 7749 


7823 


74 


586 


7898 


7972 


8046 


8120; 8194! 8268 


8342 


8416 8490 


8564 


74 


58 7 


8638 


8712 


8786 


886o| 8934! 9008 


9082 


9 1 56 9230' 9 3o3 


74 


588 


9 3 77 


945 1 


9525 9399 9673, 9746 


9820 


9894 9968! »»42 


74 


58 9 


7701 i5 


0189 


0263: o336 


0410 


0484 


o557 


o63i 07051 0778 
1367 1440' i5i4 


74 


590 


770802 


0926' 0999 


1073 


1 146 


I22o| I2g3 


74 


5 9 , 


1 58 7 


1661 


1734 


1808 


1881 


igSS 


2028 


2102' 21751 2248 


73 


592 


2322 


23 9 5 


2468 


2542 


26i5 


2688 


2762 


2835 2908 2981 


73 


5 9 3 


3o55 


3i28 


3201 


3274 


3348 


3421 


3494 


3567 364o; 37i3 


73 


5 9 4 


3 7 86 


386o 


3 9 33 


4006 


4079 


4i52 


4225 


4298 43 7 i 


4444 


73 


5g5 


45i7 


4590 


4663 


4736 


4809 


4882 


4955 


5o2^ 5 loo 


5. 7 3 


73 


5 9 6 


5246 


5319 


5392 


5465 


5538 


56 10 


5683 


5756 1 3829 


5902 


73 


5q7 


5 97 4 


6047 


6120 


6193 


6265 


6338 


641 1 


64*83 6556 


6629 


73 


5q8 


6701 


6774 


6846 


6919 


6992 


7064 


7i3 7 


7209 7282 
7934 8006 
8658 8730 


7354 


73 


5 99 


7427 


7499 


7 5 7 2 


7644 
8368 


7717 


7789 


7862 


8079 


72 


600 


778131 


8224 


8296 


8441 


85i3 


8585 


8802 


72 


601 


8874 


8947 


9019 


9091 


9i63 


9236 


9 3o8 


938o 9432 


9524 


72 


602 


9596 


9669 


9741 


9813 


9 885 


9957 


••29 


•101 • 173 


•245 


72 


6o3 


780317 


o38 9 


0461 


o533 


o6o5 


0677 


0749 


0821 o8 9 3 


0965 


72 


604 


1037 


1 109 


1 181 


1253 


i324 


1396 


1468 


i54o 161 2 


1684 


72 


6o5 


1755 


1827 


1899 


1971 


2042 


2114 


2186 


2258 2329 


2401 


72 


606 


2473 


2544 


2616 


2688 


2 7 5 9 


283 1 


2902 


2974) 3046 


3u 7 


72 


607 


3i8 9 


326o 


3332 


34o3 


3475 


3546 


36i8 


368o 3761 
44o3' 4475 


3832 


7i 


608 


3904 


3 97 5 


4046 


4118 


4189 


4261 


4332 


4546 


7i 


609 


4617 


4689 


4760 


483 1 


4902 


4974 


5o45 


5n6' 5187 


5259 


71 


610 


78533o 


54oi 


5472 


5543 


56i5 


5686 


5 7 5 7 


5828! 58 9 9 


5 97 o 


71 


611 


604 1 


6112 


6i83 


6254 


6325 


63 9 6 


6467 


6538 ! 6609 


6680 


7i 


612 


6 7 5 1 


6822 


68 9 3 


6964 


7o35 


7106 


7H7 


, 7248 7319 


7390 


7« 


6i3 


746o 


753i 


7602 


7673 


7744 


78i5 

8522 


7885 


79561 8027 
8663 8734 


8098 


7i 


614 


8168 


823 9 


83io 


838i 


845i 


85 9 3 


8804 


7> 


6i5 


88 7 5 
958i 


8946 


9016 


9087 


9157 


9228 


9299 


9369! 9440 


95io 


7i 


616 


965i 


9722 


9792 


9863 


9933 


•••4 


••74 # i44 


•2l5 


70 


6,7 


790285 


o356 


0426 


0496 


0567 


o63 7 


0707 


0778 0848 


0918 


70 


618 


0988 


1059 


1 1 29 


1199 


1269 


1 340 


1410 


1480 i55o 


1620 


70 


619 


1691 


1761 


i83i 


1901 


1971 


2041 


21 1 1 


2181 2252 


2322 


70 


620 


792392 


2462 


2532 


2602 


2672 


2742 2812 


2882 2952 


3022 


70 


621 


3092 


3i62 


323i 


33oi 


33 7 i 


3441 35 1 1 


358i 365i 


3721 


70 


622 


3790 


3 860 


3930 


4000 


4070 


4i39 ! 4209 


4279 4349 
4976 5o45 


44l8 


70 


623 


4488 


4558 


4627 


4697 


4767 


4836 ' 4906 


5n5 


70 


624 


5i85 


5254 


5324 


53 9 3 


5463 


5532 §602 


5672 5741 


58n 


70 


625 


588o 5g49 


6019 


6088 


6i58 


6227, 6297 


6366 6436 


65o5 


69 


626 


65 7 4| 6644 


6 7 i3 


6782 1 6852 


6921 6990 


7060 7 1 29 


7,98 


69 


627 


7268J 7337 


74o6 


7475 7545 


7614 1 7683 
83o5j 8374 


7752 7821 


7890 


69 


628 


79601 8029 


8098 


8467 8236 


8443 85 1 3 


8582 


69 


629 


865i 8720 


8780 
947« 


8858! 8927 


8996 9065 


9134 9203 


9272 


6 9 


63o 


799341 9400 
8300291 0098 


9547I 9°i° 


9685i 9754 


9823 9892 


9961 


69 


63 1 


0167 


o236| o3o5 


0373. 0442 


o5n' o58o 


0648 


69 


632 


0717 0786 


o854 


o 9 23j 0992 


1 06 1 1 1 29 

1747! i8i5 


1 198 1266 


i335 


69 


633 


1404 1472 


i54i 


1609I 1678 


1884 1952 


2021 


69 


634 


2089 2 1 58 


2226 


229s 1 2363 


2432; 25oo 


2568 2637 


2705 


& 


635 


2774 2842 


2910 


2979: 3o47, 3»n6 ! 3 1 84 


3252. 332i J 3389 


636 


3457 3523 


3394 


3662I 3730; 3798 3867 


3 9 35 4oo3 


4071 


68 


637 


4i3g 4208 


4276 1 


4344! 44i2 4480 4548 


.4&i6' 4685 


4753 


68 


638 


4821, 4889 


4957; 


5o25 5093 5i6r 5229 5297I 5365 


5433 


68 


63q 


55oi| 5569 


563 7 i 


5705I 5773 584i; 


5 9 o8 
6 


5976; 6044 61 1 2 


68 


.jy 


.0 | 1 


2 j 3 | 4 | 5 | 


1). 





A TABLE 


OF LOGARITHMS FROM 1 


TO 


10,000. 


n 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. ~j 


64o 


806180 


6248 


63i6 


6384 


645 1 


6519 


6587 


6655 


6723 


6790 


68 


641 


6858 


6926 


6994 


7061 


7129 


7197 


7264 


7332 


74oo 


7467 
8i43 


68 


642 


7535 


7603 


7670 


7738 


7806 


7873 


794i 


8008 


8076 


68 


643 


821 1 1 8279 


8346 


8414 


8481 


8549 


8616 


8684 


8751 


8818 


67 


644 


8886 8953 


9021 


9088 


9i56 


9223 


9290 


9358 


9425 


9492 


67 


645 


956o| 9627 


9694 


9762 


9829 


9896 


99 6 4 


••3 1 


••98 


•i65 


67 


646 


8i0233 o3oo 


o36 7 


0434 


o5oi 


0569 


o636 


0703 


0770 


0837 


67 


647 


0904' 0971 


1039 

1709 


1106 


1 173 


1240 


i3o7 


1374 


i44i 


i5o8 


67 


648 


i575 1642 


1776 


i843 


1910 


1977 


2044 


21 1 1 


2178 


67 


649 


2245 23l2 


2379 


2445 


25l2 


25 79 


2646 


2713 


2780 


2847 


67 


65o 


8l29l3 2980 


3o47 


3 1 14 


3i8i 


3247 


33i4 


338i 


3448 


35i4 


67 


65i 


358i | 3648 


3714 


3 7 8i 


3848 


3914 


3g8i 


4048 


4114 


4181 


67 


65? 


4248 


43i4 


438i 


4447 


45i4 


458 1 


4647 


4714 


4780 


4847 


67 


653 


49i3 


4980 


5o46 


5n3 


f/ 79 


5246 


53i2 


53 7 8 


5445 


55n 


66 


654 


5578 


5644 


5711 


5 777 


5843 


5910 


5976 


6042 


6109 


6173 


66 


655 


6241 63o8 


63 7 4 


644o 


65o6i 65 7 3 


663 9 


6705 


6771 


6838 


66 


656 


6904' 6970 


7o36 


7102 


7169 


^ 3 ? 


73oi 


736 7 


7433 


7499 


66 


657 


7565 7 63i 


7698 


7764 


783o 


7896 


7962 


8028 


8094 


8160 


66 


658 


8226I 8292 


8358 


8424 


8490 


8556 


8622 


8688 


8 7 54 


8820 


66 


65 9 


88851 8q5i 


9017 


9083 


9149 


92 1 5 


9281 


9346 


9412 


9478 


66 


660 


8i9544 


9610 


9676 


974i 


9807 


9873 


99 3 9 


•••4 


••70 


•i36 


66 


661 


820201 


0267 


o333 


0399 


0464 


o53o 


o5 9 5 


0661 


0727 


0792 


66 


662 


o858 


0924 


0989 


io55 


1 1 20 


1 186 


I25l 


i3i7 


i382 


1448 


66 


663 


i5i4 


1D79 

2233 


1643 


1710 


i 77 5 


1841 


1906 


1972 


2037 


2io3 


65 


664 


2168 


2299 


2364 


243o 


2495 


256o 


2626 


2691 


2756 


65 


665 


2822 


2887 


2932 


3oi8 


3o83 


3i48 


32i3 


3279 


3344 


3409 


65 


666 


3474 


3539 


36o5 


3670 


3 7 35 


38oo 


3865 


3930 


3996 


4061 


65 


667 


4126 


4191 


4256 


432i 


4386 


445 1 


4016 


458 1 


4646 


471 1 


65 


668 


4776 


4841 


4906 


4971 


5o36 


5ioi 


5i66j 523i 


5296 


536i 


65 


669 


5426 


5491 


5556 


562i 


5686 


575i 


58 1 51 588o 


5 9 45 


6010 


65 


670 


826075 


6140 


6204 


6269 


6334 


6399 


6464! 6528 


6393 


6658 


65 


671 


6723 


6787 


6852 


6917 


6981 


7046 


7111 7175 


7240 


73o5 


65 


m 


7369 
8oi3 


7434 


7499 


7563 


7628 


7692 


7757 7821 


7886 


7g5i 


65 


673 


8080 


8i44 


8209 


8273 


8338 


8402 J 8467 


853 1 


8595 


64 


674 


8660 


8724 


8789 


8853 


8918 
9361 


8982 


9046! 91 1 1 


9175 


9239 


64 


675 


93o4 


9 368 


9432 


9497 
•i3 9 


9625 


9690 9754 


9818 


9882 


64 


676 


9947 


••11 


••75 


•204 


•268 


•332 


•396 


•460 


•325 


64 


678 


830D89 


o653 


0717 


0781 


o845 


0909 

1330 


0973 


1037 
167& 


1102 


1 166 


64 


I23o 


1294 


i358 


1422 


i486 


1614 


1742 


1806 


64 


679 


1870 


1934 


1998 
2637 


2062 


2126 


2189 


2253; 23T7 


238i 


2445 


64 


680 


832D09 


2D73 


2700 


2764 


2828 


2892 2956 


3020 


3o83 


64 


681 


3i47 


321 I 


3275 


3338 


3402 


3466 


353o 3593 
4ib6' 423o 


365 7 


3721 


64 


682 


3 7 84 


3848 


3ol2 


3975 


4039 


4io3 


4294 


435 7 


64 


683 


4421 


4484 


4548 


461 1 


46 7 3- 


4739 
5373 


4802 4866 


4929 


4993 


64 


684 


5o56 


5l20 


5i83 


5247 


53ro 


5437 55oo 


5564 


5627 


63 


685 


5691 


5754 


5817 


588 1 


5 9 44 
6577 


6oot 


607 1 J 6 1 34 


6197 


6261 


63 


68b 


6324 


6387 


6401 


65u 


6641 


6704 1 6767 


683o 


6894 


63 


687 


8219 


7020 


7o83 


7U6 


7210 


7273 


7336 7399 


7462 


7525 
8i56 


63 


688 


7602 


77i5 


7778 
8408 


7841 


mt 


7967 8o3o 
85 9 7 8660 


8093 


63 


689 


8282 


8345 


8471 


8723 


8786 


63 


690 


638849 


8912 


8 97 5 


9038 


9101 


9164 


9227 9289 
9 855 9918 


9352 


94i5 


63 


691 


9478 


9341 


9604 


9667 
0294 


9729 


9792 


9981 


••43 


63 


692 


840106 


0169 


C232 


o35 7 


0420 


0482 0343' 


0608 


0671 


63 


693 


0733 


0796 


o85o 
1485 


0921 
1547 


0984 


1046 


1109J 1 172 
1735, 1797 


1234 


"97 


63 


694 


1359 


1422 


1610 


1672 


i860 


IQ22 

2547 


63 


695 


19851 2047 


2110 


2172 


2235 


2297 


236o 2422 


2484 


62 


696 


2609 26T2 


2734 


2796 


2839 


2921 


2983 3o46 


3io8 


3170 


62 


697 
698 


3233 3290 


3357 


3420 


3482 


3544 


36o6 366n 


3 7 3i 


3793 


62 


3855! 3oi8 
4477 4539 


3980 


•4042 


4104 


4166 


4229 1 4291 


4353 


44i5' 62 


699 


4601 


4664 


4726 


4788 


485o 4912 


4974 


5o36 62 


N. 


J i 


2 


_J__L 4 __ 


5 


6 ! 7 


8 


9 I D. | 



12 


A TABLE 


OF 


LOGARITHMS FROM ] 


TO 


10,000. 




N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 

62 


700 


845098 


5i6o 


5222 


5284 


5346 


5408 


5470 


5532 


55g4 


5656 


701 


5718 


5780 


5842 


5904 


5 9 66 


6028 


6090 


6i5i 


6213 


6273 


62 


702 


633 7 


6399 


6461 


6523 


6585 


6646 


6708 


6770 


6832 


6894 


62 


-ro3 


6955 


7017 


7079 


7i4i 


7202 


7264 


7326 


7 388 


7449 
8066 


761 1 


62 


704 


7373 
8189 


7634 


7696 


7738 


7819 


788! 


7943 
8559 


8004 


8128 


£1 


7o5 


82D1 


83i2 


83 7 4 


8435 


8497 


8620 


8682 


8743 


62 


706 


88o5 


8866 


8928 


8989 


905 1 


9112 


,9 X 74 


9235 


9297 


9 358 


61 


707 


9419 


948i 


9542 


9604 


9665 


9726 


9788 


9849 


991 1 


9972 
o585 


61 


708 


85oo33 


0095 


oi56 


0217 


0279 


o34o 


0401 


0462 


0324 


61 l 


709 


0646 


0707 


0769 


o83o 


0891 


0932 


1014 


1075 


u36 


1 197 


61 


710 


85i258 


1320 


i38i 


U42 


i5o3 


1 564 


i625 


1 686 


1747 


1809 


61 


711 


1870 


193 1 


1992 


2o53 


2114 


2175 


2236 


2297 


2358 


2419 


61 


712 


2480 


2541 


2602 


2663 


2724 


2 7 85 


2846 


2907 


2968 


3029 


61 


7 i3 


3090 


3i5o 


321 1 


3272 


3333 


33 9 4 


3455 


35i6 


3577 


363 7 


61 


7i4 


36 9 8 


3759 


3820 


388i 


3941 


4002 


4o63 


4124 


4i85 


4245 


61 


7 i5 


43o6 


4367 


4428 


4488 


4549 


4610 


4670 


473i 


4792 


4852 


61 ■ 


716 


4913 


4974 


5o34 


5095 


5i56 


52i6 


5277 


5337 


53 9 8 


5459 


61 


717 


5019 


558o 


564o 


5701 


5 7 6i 


5822 


5882 


5943 


6oo3 


6064 


61 


718 


6124 


6i85 


6245 


63o6 


6366 


6427 


6487 


6548 


6608 


6668 


60 


719 


6729 


6789 


685o 


6910 


6970 


703 1 


7091 


7162 


7212 


7272 


60 


720 


85 7 332 


7 3 9 3 


7453 


7 3i3 


7374 


7634 


7694 


7755 


78i5 


7875 


60 


721 


7 9 35 


7 Q9 5 


8o56 


8116 


8176 


8236 


8297 


835 7 


8417 


8477 


60 


722 


8537 


85 97 


865 7 


8718 


8778 


8838 


8898 


8g58 


9018 


9078 


60 


7 23 


9 i38 


9,98 


9258 


9 3i8 


9379 
9978 


9439 


9499 


9559 


Q619 


9679 


60 


724 


97 3 9 


9799 


9 85o 


99.8 
o5i8 


••38 


••98 


•i58 


•218 


•278 


60 


720 


86o338 


o3 9 8 


0458 


0378 


0637 


0697 


0737 


0817 


0877 


60 


726 


0937 


0996 


io56 


1116 


1 1 76 


1236 


1295 


i355 


i4i5 


U75 


60 


727 


1 534 


1594 


1 654 


17U 


1773 


1 833 


i8 9 3 
2489 


io52 

2349 


2012 


2072 


60 


728 


2l3l 


2191 


225l 


23lO 


2370 


243o 


2608 


2668I 60 


729 


2728 


2787 


2847 


2906 


2966 


3o25 


3o85 


3i44 


3204 


3263 


60 


73o 


8o3323 


3382 


3442 


35oi 


356i 


362o 


368o 


3739 


3 799 


3858 


5 9 


7 3i 


3917 


3977 


4o36 


4096 


4i55 


4214 


4274 


4333 


4392 

4g85 


4452 


5 9 


732 


431 1 


4570 


463o 


4689 


4748 


4808 


4867 


4926 


5o45 


5 9 


7 33 


5io4 


5i63 


5222 


5282 


534i 


5400 


5459 


55: 9 


5578 


563 7 


59 


7 34 


56 9 6 
6287 


5 7 55 


58i4 


58 7 4 


5 9 33 


5o 9 2 
6583 


6o5i 


6110 


6169 


6228 


5 9 


735 


6346 


64o5 


6465 


6524 


6642 


6701 


6760 


6819 


5 9 


736 


6878 


6937 


6906 


7o55 


7114 


7173 


7232 


7201 


735o 


7409 

7998 


5 9 


7 3 7 


7467 


1626 


7385 


7644 
8233 


77°3 


7762 
83 5o 


7821 


7880 


7939 


5 9 


738 


8o56 


8u5 


8174 


8292 


8409 


8468 


8527 


8586 


5 9 


7 3 9 


8644 


8703 


8762 


8821 


8879 


8 9 38 


8997 


9o56 


9u4 


1 9H3 


59 


740 


869232 


9290 


9349 


9408 


9466 


9325 


9 584 


9642 


9701 


9760 


5 9 


74i 


9818 


9877 


9935 


9994 


••53 


•in 


• 170 


•228 


•287 


•345 


s 


742 


870404 


0462 


o52i 


0379 


o638 


0696 

1281 


0755 


o8i3 


0872 


0930 
i5i5 


743 


0989 


1047 


1 106 


1 164 


1223 


1 339 


i3 9 8 


1 456 


5P 


744 


1073 


i63i 


l6qo 


1748 


1806 


i865 


i 9 23 


1981 
2564 


2040 


2098 


58 


745 


2i56 


22l5 


2273 


233i 


2389 


2448 


25o6 


2622 


2681 


58 


746 


2 7 3 9 


2797 


2855 


2913 


2972 


3o3o 


3o88 


3i46 


3204 


3262 


58 


747 


332i 


3379 3437 


3 49 5 
4076 


3553 


36u 


3669 


2727 


3 7 85 


3844 


58 


748 


3902 


3 9 6o 


4018 


4i34 


4192 


425o 


43o8 


4366 


4424 


58 


749 


4482 


4540 


45 9 8 


4656 


47U 


4772 


483o 


4888 


4Q45 
5324 


5oo3 


58 


700 


875061 


5i 19 

56 9 8 


5177 


5235 


5293 


535i 


5409 


5466 


5582 


58 


7 5i 


564o 


5756 


58i3 


58 7 i 


if 9 

6307 


5987 


6o45 


6102 


6160 


58 


702 


6218 


6276 


6333 


6391 


6449 
7026 


6564 


6622 


6680 


6 7 3 7 


58 


^ 3 


6 79 5 


6853 


6910 


6968 


7083 


7U1 


7199 


7256 


73i4 


58 


7 C 5 4 


7 3 7 i 


7429 


7487 


7344 


7602 


765 9 
8234 


77n 


7774 
8349 


7832 


7889 


58 


7 55 


8522 


8004 


8062 


8119 


8177 


8292 


8407 


8464 


?7 


7 56 


8579 863 7 


8694 


8 7 52 


8809 


8866 


8924 


8981 


9039 


P 


7 58 


9096 


9 i53! 921 1 


9268 


9325 


9383 


9440 


9497 


9555 


9612 


5 J 


9609. 9726! 9784 


9841 


9898 


q 9 56 
0328 


••i3 


••70 


•127 


•i85 


i 1 


75 9 


880242 j 0299 s °356 


o4i3 


0471 


o585 


0642 


0699 


o 7 56 


57 


N. 


| 1 j 2 


3 


4 


5 


J_ 


'- 7 


8 


_9__ 


D. 





A TABLE 


OF 


LOGARITHMS FROM 1 


to 10,000 


13 


N. 

760 





1 1 > 


I'M 


5 | 6 | 7 j 8 


9 ! I>- 


880814 


0871 0928 


0985, 1042 


1099' n56 i2i3 1271 


1328' 57 


761 


i3S5 


1442 1499 


i556 i6i3 


1670; 1727 1784 1841 


1898 


57 


762 


1955 


2012 2069 
258i| 2638 


2126: 2i83 


2240 


2297! 2354 241 1 


2468 


5 7 


763 


2523 


269a. 2752: 2809 


2866; 2923, 2980 


3o3 7 


57 


764 


3093 


3i5o 3207 


3264I 332i 1 3377 


3434 34qi 354S 


36o5 


^7 


760 


366 1 


3718 3 77 5 


3832; 3888 3^45J 4002! 40D9I 41 15 


4172 


57 


766 


4229 


4285, 4342 


4399' 4455 43i2i ^569 4625 4682 


4739 


57 


768 


4795 


4852 4909 


4965. 5o22 5078! 5i35. 5192 5248 


53o3 


57 


536 1 


5418 5474 


553i. 5587 5644' 5700 5757J L8i3 


58 7 o 


57 


769 


5926 


5q83 


6039 


6096, 6i32 6209 6265 632i! 6378 


6434 


56 


77c 


886491 


6547 


6604 


6660! 6716; 6773: 6829 


6885, 6942 


6998 


56 


771 


7004 


71 1 1 


7167 


7223; 7280 7336 7392 


7449 75o5 


756i 


56 


772 


7617 


7674 7730 


7786 7842I 7898: 7955 


801 1 8067 


8123 


56 


77 3 


8179 


8236 8292 


8348 8404 8460 8016 


8573 8629 


8685 


56 


774 


8741 


8797 8853 


8909 89651 9021 9077 


9134 9!9° 


9246 


56 


775 


9J02 


9358 


9414 


9470 9026: 9582J 9038 

••3o ••86 l »i4i # I97 


9 6g4 ! 9730 


9806 


56 


776 


9862 


9918 


9974 


•253 »3o9 


•365 


56 


777 


890421 


0477 


o533 


o589j 0645 1 070OJ 0760 


0812 0868 


0924 


56 


778 


0980 


io35 


1091 


1147J i2o3' 1259 1 i3i4 


1370 1426 


1482 


56 


779 


:53 7 


1593 


1649 


I7o5 


1760 1816 1872 


10281 1983 


2o3g 


56 


780 


892095 


2 1 DO 


2206 


2262 


23i 7 23 7 3 


24:9 2484I 254o! 25o5 


56 


781 


265 1 


2707 


2762 


2818 


2S7I 2929 


298:1 3o4oj 3096 


3i5i 


56 


782 


3207 


3262 


33i8 


33 7 3 


3429 1 3484 


33^0, 35g5 365i 


3 7 o6 


56 


783 


3762 


38i 7 


38 7 3 


3928 


3984' 4o39 


4094 


4;5o 4203 4261 


55 


784 


43i6 


43 7 i 


4427 


4482 


4338, 4593 


4648 


4704 4759 4814 


55 


785 


4870 


4925 


49^0 


5o36 


5ooi 5i46 


5201 


5257 53 1 2 


5367 


55 


786 


5423 


5478 


5533 


5588 5644 5699 


5734 


58o 9 ' 5864 


5920 


55 


787 


5 ?7 5 


6o3o 6o85 


6140 6iq5 62DI 


63o6 


636i| 6416 


6471 


55 


788 


6326 


658 1 6636 


6692 


6747, 6802 


6837 


691 2 j 6967 


7022 


55 


789 


7077 


7]32, ?I 8 7 


7242 


7297; 7332 


7407 


7462! 7317 
8012 8067 


7572 
8122 


55 


790 


897627 


7682 ( 77 3 7 
823i 8286 


7792 


7847; 79 02 


7957 


55 


791 


8176 


834i 


83o6 845i 


85o6 


856i! 86i5 


8670 


55 


792 


8^25 


8780 8835 


8890 8944; 8999 


9054 


9109 9164 


9218 


55 


7 9 3 


927,3 


9328 9 383 


9437 


9492; 9 5 47 


9602 


9656 971 1 


9766 


55 


794 


9821 


9873 9930 9985 


••3g »*94 


•i4g 


•2o3 »258 


•3l2 


55 


7 9 5 


900367 


0422 0476 


033 1 


o586 0640 0693 0749 0804 


0859 
1404 


55 


796 


0913 


0968 1022 


1077 


ii3i| u 86 


1240 1295 1349 


55 


797 


1458 


i5i3 1567 


1622 


1676 i 7 3i 


1783, 1840 1894 


1948 


54 


798 


2003 


2057 2112 


2166 


2221 2275 


232Q 2384 2438 


2492 


54 


799 


2547 


2601 


2655 


2710 2764 1 2818 


2873 2927, 2981 


3o36 


54 


800 


903090 


3 144 


3199 


3253 


33o 7 ; 336i 


34i6 3470 3524 


35 7 8 


-54 


801 


3633 


368 7 3741 


3 79 5 
4337 


3849 3904 


3958 4012 4066 


4120 


54 


802 


4i74 


4229 4283 


43gi i 4445 


4499 4553 4607 


4661 


54 


8o3 


4716 


477° 4824 


4878 


4932 4986 


5o4o 5094 5 1 4 s 


5202 


54 


804 


5256 


53 10 5364 


5418 


5472- 5526 


558o 5634 5688 3742 


54 


8o5 


5796 


585o 5904 


5g58 


6012 6066 


61 19 6173 6227 6281 


54 


806 


6335 


638 9 6443 


6497 


655 1 1 66o4 


6658 6712 6766 6820 1 54 


807 
808 


6874 


6927 6981 


7o35 70S9 7U3 


7196 7250 73o4 73581 54 


74i 1 


7465 7319 


7 5 7 3 
8110 


7626: 7680 


7734 7787 7841! 7895, 54 
8270 8324 83 7 8 843 11 54 


809 


7949 


8002 8o56 


8i63| 8217 


810 


908485 


8539 85g2 


8646 


8699: 8 7 53 


8807 | 8860 8c 14 8967! 54 


8n 


9021 


9074 9128 


9181 


9235 9289 


9342 9396 9449 93o3 34 


812 


9556 


9610 9663 


9716 


9770 9823 


9877 
0411 


993o 9984 ## 3 7 


53 


8i3 


910091 


0144 0197 


025l 


o3o4' o338 


0464 o5i8, o5 7 i 


53 


814 


0624 


0678 0731 


0784 


o838, 0891 


0944 0998 io5i: 1 1 04 


53 


8i5 


n58 


1211 


1264 


i3i7 


i3 7 i 1424 


1477, i53o i584: 1637 


53 


8i5 


1690 


1743 


I7Q7 


i85o 


1903 1936, 2009 2o63 21 16 2169 


53 


8l l 

818 


2222 


2275 


2328 


238i 


2435 2488 254i 2594; 2647 1 2700 
2066 3oiqI 3o72 ; 3i25 3178 3a3i 


53 


2 7 53 


2806 


285g 


2913 


53 


819 


3284 


333/ 


3390 


3443 34o6' 3549 36o2i 3655 3708 


3761 


53 


N. 





1 


~?~ 


3 1 4 ! 5 | 6 


_J.1. 8 ._ 


~?~[}>Z 



14 



A TABLB OF LOGARITHMS FROM 1 TO 10,0OU. 



N. 





1 


« ! 


3973 


4 | 5 


6 ! 7 


8 | 9 | 


r>. 


820 


9i38i4 


386 7 


3920I 


4026; 4079 


4i32| 4184 


4237 4290J 53 


821 


4343 


4396 


4449 45o2 


45551 4608! 4660! 4713 


4766! 4819! 53 


822 


4872 


4925 4977I 5o3o 


5o83! 5i36i 5i8g ! 524i 


52 9 4 5347 53 


823 


5400 


5453: 55o5 


5558 


56iil 5664! 5716 5769 


5822 5875 


53 


824 


5 9 2 7 


5980 1 6o33 


6o85 


6i38 6191I 6243 6296 


6349| 6401 


53 


823 


6454 


65o7i 655g 


6612 


6664' 6717! 6770 6822 


6875J 6927 


53 


826 


6980 


7o33 7083 


7i38 


71901 


7243 7295; 7348 


7400 7453 


53 


827 


75o6 


7558! 7611 


7663 


77i6, 


7768 7820' 7873 

82 9 3 8345 8397 


7925; 7978 


5 2 


828 


8o3o 


8o83: 8i35 


8188 


824o ; 


845oj 85o2 


52 


82Q 


8555 


8607. 8659 


8712 


8 7 64 ! 


8816 8869 1 8921 


8973! 9026 


52 


83o 


919078 


9i3o 9183 


9235 


9287; 


9340' 


9392! 9444 


9496; 9549 


52 


83 1 


9601 


9653 9706 


97 58 


9810 


9862! 


9914 9967 


••i 9 -.7! 


52 


832 


920123 


0176 0228 


0280 


o332 


o384 


o436 0489 


0341 1 0593 


52 


833 


o645 0697 0749 


0801 


o853 


0906 


0958' 1010 


1062 


1114 


52 


834 


1166, 1218 1270 


1322 


1374 


1426 


1478 1 i53o 


1 582 


1 634 


52 


835 


1686 


1738 1790 


1842 


1894 


1 946 


1998; 2o5o 


2102 


2 1 54 


52 


836 


2206 


2258 23l0 


2362 


24U 


2466 


23l8' 2570 


2622 


2674 


52 


837 
838 


2725 


2777 2829 


2881 


2933 2985 


3o37 3089 


3i4o 3io2 


52 


3244 


3296 334« 


3399 


345i! 35o3 


3555 36o7| 3658, 3710 


52 


83 9 


3762 


38i4 3865 


3917 


3969' 4021 


4072 4124 


4176 4228 


52 


840 


924279 


4 33 1 4383 4434 


4486 4538 


4589 4641 


4693I 4744 


52 


841 


4796 


4848 4899 
5364 54 1 5 


49^1 


5oo3 5o54 


5io6 5i5~i 


5209 1 5261 


52 


842 


53i2 


5467 


55i8 55 7 o 


5621 • 5673 5725 5776 


52 


8i3 


5828 


5S79 5 9 3i 


5982 


6o34 6o85 


6i37 6188! 6240 


6291 


5i 


844 


6342 63g4 6445 


6497 


6548 6600 


665 1 6702! 6754 


68o5 


5i 


845 


6857 1 6908 6959 


701 1 


7062 71 1 4 


7 165 7216 


7268 


7319 


5i 


846 


7 3 7°| 7422 7473 


7524 


7576 7627 


7678 77 3o 


778i 


7832 


5i 


847 


7883; 7Q 35 7986 


8o3j 


8088 8140 


9191 8242 


8293 


8345 


5i 


848 


83 9 6' 8447 8498 


8549 


8601 8652 


8703 8754 


88o5 


885 7 


5! 


849 


8908 j 8959 9010 


9061 


9112 9163 


92 1 5 9266 


9317! 9 368 


5i 


85o 


929419; 9470 9 521 


9572 


9623 96-74 


9725 9776 


9827 9879 


5i 


85i 


9930 1 g 9 8l «»32 


••83 


•i34 # i85 


•236 ^287 


•338 


•38 9 


5i 


852 


930440 


0491 o542 


0592 


0643 0694 


0745 0796 0847 


0898 


5i 


853 


0949 


1000 io5i 


1 102 


1 1 53 1 1204 


:254 i3o5 


i356 


1407 


5i 


854 


1458 


1 509 i56o 


1610 


1661 17 1 2 


1763 1814 


i865j igi5 


5i 


855 


1966 


2017 2068 


2118 


2169' 2220 


227I 2322 


2372 2423 


5i 


856 


2474 


2524 2575 


2626 


2677 


2727 


2778! 2829 

3285- 3335 


2879 1 2g3o 


5i 


857 


2981 


3o3i 3o82 


3i33 


3i83 


3234 


3386 


343 7 


5i 


858 


3487 


3538 3589 


363g 


3690 


3740 


3791] 384i 


3892 


3 9 43 


5i 


85 9 


3993 


I 4044 4094 


4145 


4195 


4246 


4296 4347 


4397 


4448 


5i 


-860 


934498 


4549 4599 


465o 


4700 


475i 


4801 4852 


4902 4953 


5o 


861 


5oo3 


! 5o54 5 1 04 


5i54 


52o5 


5255 


53o6 5356 


5406 5457 


5o 


862 


5507 


5558 56o8 


5658 


5709 


5759 


5809 586o 


5gio 5g6o 


5o 


863 


601 1 6061 61 ii 


6162 


6212 6262 


63 1 3 63631 641 3' 6463 


5o 


864 


65 14 


1 6564 6614 


6665 


6715 6765 


68 1 5 6865 6916 6966 


5o 


865 


7016 


7066 71 17 


7167 


7217: 7267 


7317: 7367 7418 7468 


5o 


866 


75i8 


7568 7618 
8069' 81 19 


7668 


7718, 7769 


7819 7869 7919 7969 


5o 


867 


8019 


8169 


8219 8269 


8320 83701 8420 


8470 


5o , 


868 


8520 


8570 8620 


8670 


8720 8770 


8820 8870 


8920 


8970 


5o 


869 


9020 


9070 9120 


9170 


9220 9270 


9320 9369 


9 4iQ 


}46q 


5o 


870 


939519 


1 9569 9619 


9669 


9719 9769 9819J 9869 


9918 5968 


5o 


871 


940018 0068 01 18 


0168 


0218 02671 °3i] 0367: 0417I 0467 
0716 0765I 0810J o865' 091 5 0964 


5o 


8,2 


o5i6 o566 0616 


0666 


5o 


873 


1014 1064 1 1 1 4 


n63 


i2i3 1263! i3i3 ! i362; 1 4-i 2 1 1462 


5o 


874 


1 5 1 1 1 56 1 161 1 


1660 


1710 1760 


1809 1859 1909 1958 


5o 


875 


200S 2o58 2107 


2157 


2207j 2256 


23o6 2355 24o5j 2455 


5o 


876 


25o4 2554 26o3 


2653 


2702I 2752 


2801 ! 285i 2901 1 2950 


5o 


^ 


3oooi 3049 3099 


3i48 3i 9 8 3247 


3297 3346 33g6J 3445 


i 5 9 


3495 3544 3590 


3643! 36o2 ! 3742 


3791! 384i 38901 3g3g 
4285 4335 43841 4433 


^ 9 


879 


3989 4o38 4o88| 4i3-»' 4186 4236 


Ls. 


. N ' 


| 1 


1 2 


1 3 | 4 j 5 


b | 7 i 8 1 9 : 


!.?•__ 



A TABLE OF LOGARITHMS FROM 1 TO 10,000. 



IS 



N. j 


! .! 


2 


3 ! 


4 


* 1 


6 1 7 


_!_[ 


9 1 


880 


9444H3 4532 


458i 


463 1 


468o : 4729' 


4779 1 4828| 


4877; 4927 1 


881 


4976, 5o2D| 3074 


5 1 24 


5l73 5222 


5272 532 1 1 5370 5419 


882 


0469! 55i8 5567 


56i6 


5665 57i5 


5764 ; 58i3| 5862 5912 


883 


0961 6010 


6059 


6to8 


6157, 6207 


6256 63o5' 6354 64o3 


884 


64D2 1 65oi 


655 i 


6600 


6649' 0DQ 8 


6747 6796: 6845, 6894 


885 


6943 6992 


7041 


7090 


7140, 7189 


7238 7287; 7336 7385 


886 


7434 ; 7483 7532 


758i 


763o 7679 
8119' 8168 


7728 7777' 7826 7875 


887 
888 


7924 : 7Q73 8022 


8070 


8217 8266 83 1 5 8364 


84i3, 


8462I 85n 


856o 


8609 8657 


8706 8 7 55, 8804 8853 


889 


8902 
949390 


895 1 8999 


9048 


9097 9146 


9195 9244 9292 9341 


890 


9439! 9488! 


9536 


9585 9634 


9683 9731, 9780 9829! 


891 


9878, 9926 9975! 


••24 


••73 •I2t 


•170 '219 ^267 »3i6 


892 


95o365- 04 14I 0462 


o5i 1 


o56o 0608 


0657 ' 0706 0754 o8o3. 


693 


o85i| 0900! 0949 


0997 


1046 1095 


1 i43 1192 1240 1289! 


894 


i338 i386 1435 


1483 


i532 i58o 


1629 16771 1726 1775I 


895 


1823I 1872 


1 92o| 1969' 


2017! 2066 


2114 2i63 2211: 2260 


896 


23o8j 2356 


24o5, 2 453 


2502 255o 


2599 2647' 2696 2744 
3o83: 3i3i! 3i8o 3228 


897 


2792I 2841 


2889 1 2933, 


2986 3o34 


898 1 


3276I 3325 


33 7 3j 342 1 


3470 


35i8 


3566 3(Si 5: 3663 3711 


899 


3 7 6c 


38o8 


3856 


3905: 


3953 


4001 


4049' 4098, 4146 4194 


900 


954243 


4291 


4339 


438 7 


4435 


4484' 


4532! 458o, 4628 4677 


901 


4725 


4773 


4821 


4869 


4918 


4966 


5oi4l 


5o62i 5no 5 1 58 


902 


6207 


5255 


53o3 


535i 


53 99 

588o 


5447 


54 9 5' 


5543 1 5592 5640 


903 


5688 


5736 


5 7 84 


5S3 2 i 


5928 


5 97 6 


6024 6072 6120 


904 


6168 6216 


6265 


63i3 


636 1 


64091 
6888 


6457 


65o5! 6553j 6601 


9o5 


6649 1 6697 


6745 


6793 


6840 


6 9 36 


6984! 7032 i 7080 


906 


71281 7176 


7224 


7272 


7320 


7368 


74i6 


7464', 75i2; 7559 


907 


7607 ! 7655 


77 o3 


775i 


7799 


7847 


7894 


7942 


7990 8o38 


908 


8086 j 8 1 34 


8181 


8229 


8277 


8325 


83 7 3 


8421 


8468-, 85 1 6 


909 


8564 8612 


865g 


8707 


8 7 55 


88o3 


885o 


8898 


8946 1 8994 


910 


959041 1 9089 


9137 


9 i85 


9232 


9280 


9328 


93]5 9423 9471 


911 


95i8i 9566 


9614 


9661 


9709 


97D7 


9804 


9852! 9900' 9947 


912 


9995 


••42 


••90 


•i38 


•i85 


•233 


•280 


•328 


•376, »423 


913 


96047 1 


o5i8 


o566 


o6i3 


0661 


0709 


0756 


0804 


o85i 0899 


914 


0946 


0994 


1041 


1089 


i/36 


1 184 


I 23 1 


1279 


i326 ! 1374 


gi5 


1421 


1469 


i5i6 


1 563 


161 1 


1658 


1706 


1753 


1801 j 1848 


916 


i8 9 5 


1943 


1990 


2038 


2oS5 


2l32 


2180 


2227 


2275i 2322 


91 I 


2369 


2417 


2464 


25ll 


2559 


2606 


2653 


2701 


2748| 2 79 5 


918 


2843 


2890 


2937 


2985 


3o32 


3079 


3i26 


3n4 


322I 1 3268 


919 


33i6 


3363 


34io 


3457 


35o4 


3552 


35 9 9 


3646 


36 9 3; 3741 


920 


963788 


3835 


3882 


3929 


3977 


4024 


4071 


4118 


4i65 4212 


921 


4260 


43o7 


4354 


1 4401 


4448 


4495 


4542 


45go 


4637 


4684 


922 


473i 


4778 


4825 


4872 


4919 


! 4966 


5oi3 


5o6i 


5 1 08 


5i55 


923 


5202 


5249 


5296 


5343 


53 9 o 


! 5437 


5484 


553 1 


5578 


5625 


924 


5672 


5719 


5766 


58 1 3 


5S6o 5907 


5 9 54 


6001 


6048 


6og5 


926 


6l42 


6189 
6658 


6236 


6283 


6329! 6J76 


6423 


6470 


65i 7 


6564 


926 


661 1 


6700 


6752 


6799 ! 6845 


6892 


6939 


6986 


7 o33 


927 


7080 


7127 


7173 


7220 


7267| 73i4 


736i 


7408 


7454 


75oi 


928 


7548 


7595 


7642 


7688 


77 35| 7782- 


7829 


7 8 7 5 7922 
8343 83go 


7069 


929 


8016 


8062 


8109 


8i56 


82o3i 8249 


8296 


8436 


93o 


968483 


853o 


8576 


8623 


8670 8716 


8763 


8810 


8856 


8903 


93i 


8950 


8996 


904^ 


9090 


9i36 9183 


9229 
9 6 9 5 


9276 


g323 


9369 


932 


94l6 


9 463 


95oc 


9556 


9602 9649 


9742 


9789 


9 833 


933 


9882' 9918 997: 


••21 


••681 •uA 


•161 


•207 


•254: - 3oo 


934 


970347; o3 9 3 044c 


0486 


o533J 0579 


0626 


0672 


0719 0765 


9 35 


0812 o858| 090; 


k 0931 


0997 i 1044 


1090 


u37 


u83 1229 


9 36 


1276 i322i i36c 


> I4i5 


1461 j i5o8 


1 1 534 


1601 


1647 1 6^3 

SIIOJ 2137 


III 


1740' 1786 i83s 


1879 


1925 1971 
2383 2434 


2018 


2064 


22o3 2249 2295, 2342 


2481 


2527J 2573 ; 2619 


9 3g 


2666! 2712! 2758) 280/ 


! 285 1 i 2897 


L294 3 


2989, 3o35 3082 


1 *. 


O 


1 «■ 


1 2 


! 3 


! 4 


i s 


| 6 


rr 


! 8 


i 9 



D. 

49 
49 
49 
49 
49 
49 
49 
49 
49 
49 
49 
49 
49 



48 
48 
48 
48 
48 
48 
48 
48 
48 
48 
48 
48 
48 
48 
48 
48 
48 
47 
47 
47 
47 
47 
47 
47 

47 

47 
47 
47 
47 
47 
47 
47 

47 
47 
47 
47 
47 

46 
46 
46 
46 
46 
46 

IX 



10 


A TABLE 


OF 


LOGARITHMS FROM 1 


TO 


10,000. 




N. 
940 





1 


2 


3 


4 


5 


6 


7 


8 


9 


'DTI 

46 


973128 


3i74 


3220 


3266 


33i3 


335 9 


34o5 


345 1 


3497 


3543 


941 


3D90 


3636 


3682 


3728 


3774 


3820 


3866 


3913 


3909 


400 5 


46 


942 


4o5i 


a 


4143 


4189 


4235 4281 


4327 


4374 


4420 


4466 


46 


943 


45i2 


4604 


465o 


4696; 4742 


4788 


4834 


4880 


4926 


46 


y44 


4972 


5oi8 


5o64 


5no 5i56j 5202 


5248 


5294 
5 7 53 


534o 


5386 


46 


945 


5432 


5478 


5524 


5570 56i6| 5662 


5707 


5799 
6258 


5845 


46 


946 


5891 
635o 


5o3 7 
63 9 6 
6854 


5 9 83 


6029 


6075 6121 


6167 
6625 


6212 


63o4 


46 


947 


6442 


6488 


6533 65 79 


6671 


6717 


6 7 63 


46 


948 


6808 


6900 


6946 


6992 7037 


7083 


7129 


7175 


7220 


46 


949 


7266 


7312 


7 358 


74o3 


7449 7495 


754i 


7586 


7632 
8089 


7678 


46 


95o 


977724 


7769 
8226 


7 8i5 
8272 


7861 


7906 7952 
8363! 8409 


8454 


8o43 


8i35 


46 


95i 


8181 


83i 7 


85oo 


8546 


85 9 i 


46 


952 


8637 


8683 


8728 8774 


8819 1 8865 
9275, 9321 


891 1 


8 9 56 


9002 


9047 


46 


9 53 


9093 


9i38 


9184I 9230 


9366 


9412 


9457 


95o3 


46 


954 


9048 


9594 


9639' 9685 


973o, 9776 


9821 


9867 


&; 


99 58 


46 


955 


980003 


0049 


0094 


0140 


01 85] 023 1 
0640' o685 


0276 


0322 


0412 


45 


9D6 


0458 


o5o3 


o549 


o5g4 


0730 


O776 


0821 


0867 


45 


9D7 


0912 


0957 


ioo3 


1048 


io 9 3 n3g 


1 184 


I229 


I2ff5 


l320 


45 


9 58 


1 366 


i4n 


1 456 


i5oi 


i547 ( 1592 1637 


1 683 


1728 


1773 


45 


9 5 9 


1819 


1864 


1909 


i 9 54 


2000, 2o45i 2090 


2i35 


2181 


2226 


45 


960 


982271 


23i6 


2362 


2407 


2452 j 2497 2543 


2588 


2633 


2678 


45 


961 


2723 


2769 


2814 285 9 


2904 2949 2994 


3o4o 


3o85 


3i3o 


45 


962 


3i75 


3220 


3265 33io 


3356, 34oi 


3446 


3491 


3536 


358i 


45 


963 


3626 


36 7 i 


3716 


3762 


3807! 3852 


38 97 


3942 
4392 


3987 


4o32 


45 


| 964 


4077 


4122 


4167 


4212 


4257 43o2 


4347 


4437 


4482 


45 


! 965 


4527 


4572 


4617 


4662 


4707 | 4752 


4797 


4842 


4887 


4g32 


45 


966 


4977 


5o22 


5067 


5i 12 


5lD7 5202 


5247 


5292 


533 7 


5382 


45 


967 
968 


5426 


5471 


55i6 


556i 


56o6 565 1 


5696 


5741 


5 7 86 


583o 


45 


5875 


5920 


5965 6010 


6o55 ( 6100 


6144 


6189 


6234 


6279 


45 


969 


6324 6369 


64i 3 6458 


65o3 


6548 


65 9 3 


6637 


6682 


6727 


45 


970 


986772 6817 


6861 


6906 


6951 


6996 


7040 


7o85 


7i3o 


7175 


45 


97' 


7219 7264 


7 3o 9 


7353 


7 3 9 8 


7443 


7488 


7532 


7577 


7622 


4^ 


973 


7666 
8ii3 


8i5 7 


77 56 
8202 


7800 
8247 


7845 
8291 


7890 
8336 


it 


8425 


8024 8068 
8470 85i4 


43 
45 


974 


855o 
9005 


8604 


8648 8693 
9094 91 38 
9539 g583 


8 7 3 7 


8782 


8826 


8871 


8916 8960 


45 


973 


9049 


9i83 9227 


9272 


93i6 


9361 


94o5 


45 


976 


945o 


9494 


9628 9672 


9717 


9761 


9806 


985o 


44 


977 
978 


9895 


9939 9983 


••28 


••72 »ii7 


•161 


•206 


•25o 


•294 


44 


990339 o3b3 


0428 


0472 


o5i6 o56i 


o6o5 


o65o 


0694 


0738 


44 


979 


0783 


0827 


0871 


0916 
1359 


0960, 1004 


1049 


1093 


1 137 


1182 


44 


980 


991226 


1270 


i3i5 


Uo3 1448 


1492 


1 536 


i58o 


l6 2 5 


44 


981 


1669 


1713 


1758 


1802 


1846, 1890 


1935 


1979 


2023 


2067 


44 


982 


2 1 1 1 


2i56 


2200 


2244 


2288J 2333 


2377 


2421 


2465 


25og 


44 


9 83 


. 2554 2598 


2642 


2686 


2 7 3o 2774 


2819 


2863 


2907 
3348 


2g5i 


44 


984 


2995 3o3g 


3o83 


3127 


3172' 3216 


3260 


33o4 


33 9 2 


44 


9 85 


3436 


3480 


3524 


3568 


36i3 3657 3701 


3 7 45 


3789 


3833 


44 


986 


38 7 7 


3g2i 


3965 


4009 


4o53i 4097 
4493 4537 


4Ui 


4i85 


4229 


42 7 3 


44 


987 


43i7 


436 1 


44o5 


4449 


458 1 


4625 


4669 47 1 3 
5io8i 5i52 


44 


988 


4757 


4801 


4845 


4889 


4933, 4977 


5021 


5o65 


44 


989 


5196 5240 


5284 


5328 


53 7 2 


5416 


5460 


55o4 


5547I 5591 
5986) 6o3o 


44 


990 


9956351 5679 


5723 


5767 


58n 


5854 


58 9 8 


5 9 42 
638o 


44 


991 


6074 6117 6161 


62o5 


6249 


6293 


6337 


6424! 6468 


44 


992 


65i2 6555 6599 


6643 


6687 


6 7 3 1 


6774 


6818 


6862 6906 


44 


993 


6949 1 6993 7037 
7386 743oj 7474 


7080 


7124 


7168 


7212 


7255 


7209 
7736 


7343 


44 


994 


7 5i 7 


756i 


76o5 


7648 
8o85 


7692 
8129 


7779 
8216 


44 


99 5 


7823 7867! 7910 
8259 83o3; 8347 
86 9 5 8 7 3 9 ! 8782 
9i3i 9174 9218 


79D4 
83 9 o 


7998] 8041 
8434 8477 


8172 


44 


996 


852i 


8564 


8608 8652 


44 


997 


8826 


8869 89 1 3 8 9 56 
93o5; 9348 9392 


9000 


9043 9087 


44 


998 


9261 


9435 


9479 9522 
99 1 3 99 5 7 


44 


999 
N. 


9565 9609 9652 


9696 


9739^ 9783| 9826 


9870 


43 


1 


2 


3 


4 L 5 _ L 6 i ?._. 


8 9 ! D. 



£ TABLE 



OF 



LOGARITHMIC 
SINES AND TANGENTS 



FOR EVERY 



DEGREE AND MINUTE 
OF THE UUADKANT. 



Kemark. The minutes in the left-hand column of 
3ach page, increasing downwards, belong to the de- 
grees at the top ; and those increasing upwards, in the 
right-hand column, belong to the degrees below 






18 


(0 


DEGREES.) A TABLE 


OF LOGARITHMIC 







Sine 

0- 000000 


D. 


Cosine \ D. 


Tang. 


D. 


Cotang. 






10-000000 


0' 000000 




Infinite. 


60 


I 


6-463726 


5017.17 


000000 -00 


6-463796 


5017 


H 


i3 • 53627 i 


5 9 


2 


764756 


2934- 


85 


000000 1 "00 


764756 


2934 


83 


235244 


58 


3 


940847 


2082- 


3i 


000000 1 '00 


940847 


2082 


3i 


o59i53 


57 


4 


7.065786 


i6i5- 


J 7 


OOOOOOI '00 


7.065786 


i6i5 


17 


12-934214 


56 


5 


162696 


i3io. 


68 


OOOOOO "OO 


162696 


i3io 
1 1 1 5 


11 


83.7304 
758122 


55 


6 


241877 


11 15- 


75 


9.999999 -01 


241878 


54 


I 


308824 


966. 


53 


999999 .01 


3o8825 


852 


691 175 
633 1 83 


53 


3668i6 


852- 


54 


999999 -01 


366817 


54 


52 


9 


417968 


762- 


63 


999999 -oi 


41797° 


762 


63 


582o3o 


5i 


10 


463725 


689. 


88 


999998 


•01 


463727 


689 


88 


536273 

12-494880 

427091 


5o 


1 1 


7-5o5ii8 


629- 


81 


9.999998 


•01 


7«5o5i2o 


629 


81 


% 


12 


542906 


579. 


36 


999997 


•OI 


542909 


5 J? 


33 


i3 


577668 


536- 


4i 


999997 


•01 


577672 


536 


42 


422328 


47 


14 


609853 


499- 


38 


999996 


•01 


609857 


499 


s 


390143 


46 


i5 


63 9 8i6 


467- 
438- 


U 


999996 


•01 


639820 


467 
438 


36oi8c 


45 


16 


667845 


81 


999995 

999995 


•01 


667849 


82 


332 1 5i | 44 


J2 


694173 


41 3- 


72 


•01 


694179 


4i3 


73 


3o582i 43 


718997 


391 • 


33 


999994 


•01 


719004 


391 


36 


280997! 42 
257516' 41 


19 


742477 


3 7 i- 


27 


999993 


•01 


742484 


3 7 i 


28 


20 


764754 


353 


i5 


999993 


■01 


764761 


35i 


36 


235239 40 


21 


7-785943 
806146 


336- 


72 


9-999992 


•01 


7-780951 
8061 55 


336 


73 


12- 214049! 39 


22 


321- 


t 


999991 


•01 


321 


76 


193845 


38 


23 


82545i 


3o8 


999990 


•01 


825460 


3o8 


06 


174540 


37 


24 


843934 


295 


47 


999989 
999988 


•02 


843944 


2q5 


49 


i56o56 


36 


25 


861662 


283 


88 


•02 


861674 


263 


90 


138326 


35 


26 


878695 


2 7 3 


H 


999988 


•02 


878708 


2 7 3 


.8 


1 21292 


34 


27 


895085 


263 


23 


999987 


•02 


895099 


263 


2D 


1 0490 1 


33 


28 


910879 
926119 


253 


99 


999986 


•02 


91 0894 


254 


01 


089106 


32 


29 


245 


3S 


999985 


•02 


926134 


245 


40 


073866 


3i 


3o 


940842 


23 7 


33 


999983 


•02 


940858 


23 7 


35 


059142 


3o 


3i 


7.955082 


229 


80 


9-9999 82 , '° 2 


7 -955 100 


229 


81 


1 2 • 044900 


2 


32 


968870 


222 


73 


999981 


•02 


968889 


222 


75 


o3 1 1 1 1 


33 


982233 


216 


08 


999980 


•02 


982253 


216 


10 


017747 


27 


34 


995198 
8-007787 


209 


81 


999979 


•02 


995219 


209 


83 


004781 


26 


35 


2o3 


90 


999977 


•02 


8-007809 
020045 


203 


92 


11-992191 

979965 


25 


36 


020021 


198 


3i 


999976 


•02 


198 


33 


24 


37 


031919 
043 5o 1 


io3 


02 


999975 


•02 


031945 


193 

188 


o5 


9 68o55 


23 


38 


188 


01 


999973 


•02 


043527 


o3 


956473 


22 


39 


054781 


i83 


25 


999972 


•02 


054809 


1 83 


27 


945191 


21 


40 


. 065776 


178 


72 


999971 


•02 


o658o6 


178 


74 


934194 


20 


41 


8-076500 


174 


41 


9-999969 


•02 


8-07653i 


'74 


44 


11-923469 
9i3oo3 


:? 


42 


086965 


170 


3 1 


999968 


•02 


086997 


170 


34 


43 


097 1 83 


166 


39 


999966 


•02 


097217 


166 


42 


902783 


n 


44 


107167 


162 


65 


999964 


•03 


107202 


162 


68 


883o37 


16 


45 


1 16926 


1 5 9 


08 


999963 


•03 


1 16963 


159 
i55 


10 


i5 


46 


1 2647 1 


1 55 


66 


999961 


•o3 


i265io 


68 


873490 


14 


47 


i358io 


l52 


38 


999959 


-o3 


i3585i 


l52 


41 


864149 


i3 


48 


144953 


149 


24 


9 999 58 


-o3 


144996 
i53 9 5 2 


149 


27 


855oo4 


12 


49 


153907 


146 


22 


999956 


• o3 


146 


•27 


846048 


11 


5o 


162681 


143 


33 


999954 


• o3 


162727 


143 36 


837273 


10 


5i 


8-171280 


140 


54 


9-999952 


• o3 


8-171328 


140 57 


11-828672 


I 


52 


179713 


1 3 7 


86 


9999 5 ° 


• o3 


179763 
i88o36 


137.90 
135-32 


820237 


53 


187985 


i35 


29 


999948 


• o3 


811964 


7 


54 


196102 


1 132 


80 


999946 


• o3 


1 961 56 


132-84 


8o3844 


6 


55 


204070 


i3o 


41 


999944 -o3 


204126 


i3o«44 


795874 
788047 


5 


56 


2ii8 9 5 


1 128 


10 


999942 -o4 


21 1953 


128-14 


4 


11 


219581 


1 125 


•3 7 


999940 -04 


219641 


125-90 


780359 
772800 


3 


227134 


123 


•72 


999938, >o4 


227195 


123.76 


2 


5 9 


234557 


j 121 


■64 


999936 -o4 


234621 


121-68 


765379 


r 


60 


24i855 


1,9 


•63 


999934! -o4 


1 241921 
i Cotang. 


119-67 


758079 







Cosine 


1 *>• 


Sine |89° 


D. 




Tang. 


M. 





i? 


rXES AND TANGENTS 


(1 DEGREE.) 




1 


o 


Sine 


D. , 


Cosine | D. 


Tung. 


D. 


Cotang. | 


8 "24i855 


119-63 


9. 999934 1 


.04 


8-241921 


119 


67 


11-758079 60 


i 


249033 


117-68 


999932 j 


• 04 


249102 


H7 


72 


7 50898 I 5g 

743835 58 


a 


256094 


u5-8o 


999929; 


• 04 


256i65 


113 


84 


3 


263042 


113-98 


9999 2 7| 


• 04 


263 1 1 5 


H4 


02 


736885 37 


4 


269881 


112-21 


999923) 


•04 


269936 


112 


25 


730044 56 


5 


276614 


110-30 


999922I 


• 04 


276691 


I 10 


54 


723309; 55 


6 


283243 


io8-83 


999920 


• 04 


283323 


10b 


87 


716677 54 


2 


289773 


107-21 


999918J 


• 04 


289856 


IO7 


26 


710144I 53 


296207 


io5-65 


999913; 


•04 


296292 


io5 


70 


703708 52 


9 


302546 


io4-i3 


99991 3 


.04 


3o2634 


104 


18 


6 9 7366| 5i 


to 


308794 


102-66 


999910; 


• 04 


3o8884 


102 


70 


691 1 16 30 


is 


8- 3 14904 


101 -22 


9-999907; 


• 04 


8-3i5o46 


101 


26 


II-684954I 4Q 


13 


321027 


99-82 


999903; 


.04 


321 122 


99 


87 


678878 48 


IJ 


327016 


98-47 


999902; 


• 04 


3271 14 


98 


5i 


672886 47 


14 


332924 


97-14 


999899! 


•o5 


333o25 


97 


19 


666 97 5l 46 


i5 


33o>753 


9 5 -86 


999897 | 


•o5 


338856 


9 5 


90 


661144I 45 


16 


344504 


94-60 


999894! 


• o5 


344610 


94 


63 


6553 9 o 44 


\l 


35oi8i 


93-38 


999891! 


•03 


350289 


9 3 


43 


6497 1 1 


43 


355783 


92-19 
91 -o3 


999888 


•o5 


355893 
36i43o 


92 


24 


644 1 o5 


42 


x 9 


36i3i5 


999885 


•o5 


1 


08 


638370 


41 


20 


366777 


89-90 
88-80 


999882 


•o5 


3668 9 5 


£ 


633io5 


40 


21 


8-372171 


9.999879 


• o5 


8-372292 


11-627708 3o 
622378 38 


22 


377499 


87-72 


9998761 


•03 


377622 


37 


77 


23 


382762 


86-67 


999873 


•o5 


382889 


86 


72 


617m 


37 


24 


387962 


85-64 


999870 


• o5 


388o 9 2 
393234 


85 


70 


611908 


36 


25 


393101 


84-64 


999867; 


•o5 


84 


70 


606766 


35 


26 


398179 


83-66 


999864 


• o5 


3 9 83i5 


83 


7i 


6oi685 


34 


2 7 


4o3 1 99 


82-71 


999861 


•03 


4o3338 


82 


76 


596662 


33 


28 


408161 


8o-86 


999858 


"•o5 


4o83o4 


81 


82 


591696 


32 


2q 


4i3o68 


999854 


• o5 


4i32i3 


80 


9i 


586787 


3i 


3o 


417919 


79.96 


999851 


.06 


418068 


80 


02 


58i 9 32 


3o 


3i 


8-422717 


79-09 
78. 23 


9.999848 


.06 


8-422869 
427618 


$ 


14 


n .577131 


3 


32 


427462 


999844 


.06 


3o 


572382 


33 


432i56 


77.40 
76.57 


999841 


•06 


4323i5 


77 


45 


567685 


27 


34 


4368oo 


999838 


• 06 


436962 


76 


63 


563c38 


26 


35 


44i394 


75-77 


999834 


.06 


44i36o 


75 


83 


558440 


25 


36 


445941 


74-99 


99983 1 


-06 


446110 


75 


o5 


553890 
549387 


24 


37 


45o44o 


74-22 


999827 


-06 


45o6i3 


74 


28 


23 


38 


454893 


73-46 


999823 


• 06 


455070 


73 


52 


544930 


22 


3 9 


45o3oi 
463665 


72.73 


999820 


■ 06 


45q48i 


72 


79 


540319 


21 


4o 


72-00 


999816, 


.06 


463849 
8-468172 


72 


06 


536i5i 


20 


4i 


8-467985 


71-29 


9.999812 


.06 


7i 


35 


Il-53i828 


19 


42 


472263 


70-60 


999809 


.06 


472454 


70 


66 


527546 18 


43 


476498 


69-91 


999803 


.06 


476693 


69 


98 


5233o7| 17 


44 


480693 


69-24 


999801 


.06 


480892 


69 
6$ 


3i 


519108 


16 


45 


484848 


68-5 9 


999797 


•07 


485o5o 


65 


5i495o 


i5 


46 


488963 


67.94 
67.31 


999793 


•07 


489170 
49325o 


68 


01 


5io83o 


14 


47 


493040 


999790 
999786 


.07 


67 


38 


506750 


i3 


48 


497078 


66-69 
66-08 


.07 


497293 


66 


76 


502707 


12 


49 


5oio8o 


999782 


•07 


501298 


66 


i5 


498702 


11 


5o 


5o5o45 


65-48 


999778; 


•07 


5o5267 


65 


55 


494733 


10 


5i 


8-508974 
til 2867 


64-89 


9.999774 


•07 


8 -509200 


64 


96 


11-490800 


i 


52 


64-3i 


999769 


•07 


513098 


64 


39 


486902 


53 


516726 

52o55i 


63- 7 5 


999765 


•07 


516961 


63 


82 


483o39 


7 


54 


63-i 9 


999761 


•07 


520790 
524586 


63 


26 


479210 
475414 


6 


55 


524343 


62-64 


999757 


•07 


62 


72 


5 


56 


528102 


62-11 


999753 j 


.07 


528349 


62 


18 


47i65i 


4 


n 


53i828 


6i-58 


999748; 


•07 


532o8o 


61 


65 


467920 


3 


535523 


61-06 


999744; 


•07 


535779 


61 


i3 


464221 


2 


5 9 


539186 


60 -55 


9997- 40 ! 


•07 


53 9 447 
543o84 


60 


62 


;6o553 


; 


6o 


542819 


60-04 


999735 1 


•07 


60 


12 


456916! 




Cosino 


D. 


Sine | 


38° 


Cotang. 


D. " 


Tang~l 



20 


(2 


DEGREES.) A TABLE OF LOGARITHMIC 




M. , 


Sine 


D. 


Cosine 


D. 


Tang. 1 


D. 


Cotang. 
11 -456916 


60 





8-542819 


60 -04 


9-999735 


.07 


8-543o84l 


60 • 12 


i 


546422 


5 9 - 


55 


999731 


■07 


546691 


5 9 


62 


453309 


u 


2 


549995 
553539 


5 9 


06 


999726 -07 


500268 


u 


14 


449732 


3 


58 


58 


999722J -08 


5538i7 


66 


446i83 


57 


4 


557054 


58 


11 


999717' -08 


55 7 336 


58 


l 9 


44 2 664 : 56 


5 


56o54o 


57 


65 


999713 -08 


560828 


5 7 


73 


439172 55 


6 


563999 


^ 


l 9 


999708' -08 


564291 


57 


27 


430709! 54 
432273; 53 


I 


56743i 


56 


74 


9991041 *o8 
999699 -08 


567727 


56 


82 


570836 


56 


3o 


571137 


56 


38 


428863| 52 


9 


574214 


55 


87 


999694 -08 
999689 J -08 


574520 


55 


£ 


425480] 5 1 


10 


577066 


55 


44 


577877 
8-58i2o8 


55 


422I23i 5o 


II 


8.580892 


55 


02 


9-9996801 -08 


55 


IC 


11-418792 


% 


12 


584193 


54 


60 


999680! -08 


5845 1 4 


54 


68 


4i5486 


i3 


587469 


54 


*9 


999675! -08 


587795 


54 


27 


412205 


% 


14 


590721 


53 


79 


999670' -08 


59 1 00 1 


53 


87 


408949 


i5 


5;3 9 48 


53 


3 9 


999665J .08 


094283 


53 


il 


405717 
402 5o8 


45 


16 


597102 


53 


00 


999660! -08 


597492 


53 


44 


\l 


6oo33a 


52 


61 


999655J -o8 


6oo6tj 


52 


70 


399323 


43 


6o348<j 


52 


23 


99</j5o| -08 


6o383 9 


52 


32 


396161 


42 


19 


60662J 


5i 


86 


999645' 09 


606978 


5i 


n 


393022 


4i 


20 


609734 


5i 


49 


999640 


09 


6 1 0094 


5i 


389906 


40 


21 


8.612823 


5i 


12 


9-999635 


.09 


8 • 6 1 3 1 89 


5i 


21 


11.386811 


il 


22 


610891 


5o 


76 


999629 


-09 


616262 


5o 


85 


383738 


23 


618937 


5o 


41 


999624 


.09 


619313 


5o 


5o 


380687 


3 7 


24 


621962 


5o 


06 


999619 


• 09 


622343 


5o 


i5 


377657 

374648 


36 


25 


624965 


49 


72 


999614 


•09 


625352 


49 


81 


35 


26 


627948 


49 


38 


999608 


.09 


628340 


49 


47 


371660 


34 


27 


63oqi 1 


% 


04 


999603 -09 


63i3o8 


49 


i3 


3686 9 2 


33 


28 


633854 


71 


999597 


•09 


634256 


48 


80 


365744 


32 


29 


636776 


48 


3 9 


999562 
999586 


-09 


637184 


48 


48 


362816 


3i 


3o 


63 9 68o 


48 


06 


.09 


64ooo3 
8-642982 


48 


16 


359907 
11-357018 


3o 


3i 


8-642563 


47 


75 


9-999581 


-09 


47 


84 


29 


32 


643428 


47 


43 


999575 


.09 


645853 


47 


53 


354147 
351296 


28 


33 


648274 


47 


12 


999570 


.09 


648704 


' 47 


22 


27 


34 


65i 102 


46 


82 


99q564 


.09 


65i53 7 


46 


9 1 


348463 


26 


35 


60391 1 


46 


52 


999558 


• 10 


654352 


46 


61 


345648 25 


36 


656702 


46 


22 


999553 -io 


657149 
659928 


46 


3i 


34285i 


24 


u 


609475 


45 


92 


999547 


• 10 


46 


02 


340072 


23 


66223o 


45 


63 


999041 


• 10 


662689 
665433 


45 


•73 


3373ii 


22 


39 


664968 


45 


35 


999535 


• 10 


45 


•44 


334567 


21 


.40 


667689 


45 


06 


999529 


• 10 


668160 


45 


-26 


33 1840 


20 


41 


8-670393 


44 


•79 


9-999524 


• 10 


8-670870 


44 


•88 


11 -329i3o 


3 


42 


6730S0 


44 


5i 


999518 


• 10 


673563 


44 


• 61 


326437 


43 


675751 


44 


24 


999512 


•10 


676239 


44 


•34 


323761 


17 


44 


678405 


43 


97 


999506 


• 10 


678900 


44 


•17 


32MOO 


16 


45 


68io43 


43 


7° 


999500 


•10 


681044 


43 


-8o 


3 i 8456 


i5 


46 


683665 


43 


•44 


999493 


•10 


684172 


43 


•54 


3 1 5828 


14 


47 


686272 


43 


18 


999487 


• 10 


686784 


43 


• 28 


3i32i6 


i3 


48 


688863 


42 


92 


99948i 


•10 


68 9 38i 


43 


o3 


310619 


12 


49 


691438 


42 


67 


999475 


•10 


691963 
694029 


42 


77 


3o8o37 


11 


5o 


6 9 3 99 8 
8-696043 


42 


42 


999469 


•10 


42 


52 


3o547i 


10 


5i 


42 


17 


9-999463 


•11 


8-697081 


42 


28 


11 -302919 


I 


52 


699073 


4i 


92 


999456 


•11 


699617 


42 


o3 


3oo383 


53 


701589 


4i 


68 


999450 


•1 1 


702139 


41 


79 


297861 


I 


54 


704090 


4i 


44 


999-443 


•11 


704646 


41 


55 


295354 


55 


706577 


4i 


21 


999437 


•1 1 


707140 


4i 


32 


292860 


5 


56 


709049 


4o 


97 


99943 1 


• 11 


709618 


41 


08 


290382 


4 


n 


71 i5o7 


40 


74 


999424 


•11 


712083 


4o 


85 


287917 


3 


713952 


4o 


5i 


999418 


•1 1 


7U534 


40 


62 


285465 


2 


59 


7i6383 


4o 


29 


99941 1 


• 11 


716972 


40 


4o 


283028 


i 


60 


718800 


40 06 


999404 • 1 1 


719396 


40-17 


280604 







Cosine 


~T> 




Sine [8T° 


Cotai tg. 


D. 


Tang. 


M. 





SINES 


AND TANGENTfc 


^3 degrees/ 




21 


nsr 


Sine 


I D. 


Cosine 


D. 


T;mg. 


D. 


Cotang. j 





8-718800 


4o-o6 


9-999404 


-ii 


8-719396 


40-17 


1 1 • 280604 60 


I 


721204 


3 9 


84 


999398 


• 11 


721806 


39-95 


278194 5o 
2 7 5 79 6: 58 


2 


723595 


3 9 


62 


9 99 3oi 


• 11 


724204 


3g -74 


3 


720972 


3 9 


41 


999384 


• 1 1 


726588 


39-52 


273412 57 


4 


728337 


39 


10 


999378 


• ii 


728959 


39 -3o 


271041I 56 


5 


730688 


38 


98 


999371 


• ii 


73i3i 7 


39-09 


268683 55 


6 


733027 


38 


77 


999364 


•12 


733663 


38- 89 
38-68 


266337, 54 


7 


735354 


38 


57 


999357 


•12 


735996 


264004 53 


8 


737667 


38 


36 


99935o 


•12 


7383i7 


38-48 


26l683 52 


9 


739969 


38 


16 


999343 
999336 


•12 


740626 


38-27 


259374 5i 


10 


742239 


37 


96 


•12 


742922 


38-o 7 


257078 5o 


ii 


8-744536 


3? 


76 


9-999329 


•12 


8-745207 


3 7 -8 7 


11-254793 


4 2 


12 


746802 


37 


56 


999322 


•12 


747479 


3 7 -68 


252521 


48 


i3 


749055 


37 


37 


9993 1 5 


•12 


749740 


37-49 


25o26o 


47 


14 


751297 


37 


n 


999308 


•12 


751989 


37-29 


24801 1 


46 


i5 


753528 


36 


98 


999301 


-12 


754227 


3-7-10 


^45773 


45 


16 


755747 


36 


79 


999294 


•12 


736403 


36-92 


243547 


44 


17 


75 79 n5 


36 


61 


999286 


•12 


758668 


36-73 


24i332 


43 


18 


760101 


36 


42 


999279 


• 12 


760872 


36-55 


239128 


42 


19 


762337 


36 


24 


999272 


•12 


763o65 


36-36 


236q35 


4i 


20 


7645 11 


36 


06 


999260 


• 12 


765246 


36-i8 


234754 


4o 


21 


8- 7 e66 7 5 


35 


88 


9-999257 


•12 


8-767417 


36-oo 


n-232583 


3 9 


22 


768828 


35 


70 


999200 


•13 


769378 


35-83 


23o422 38 


23 


770970 


35 


53 


999 2 42 


•13 


771727 


35-65 


228273 


37 


24 


773ioi 


35 


35 


999235 


•13 


773866 


35-48 


226i34 


36 


25 


773223 


35 


18 


999227 


-13 


775 99 5 


35-3i 


224oo5 


35 


26 


777333 


35 


01 


999220 


•i3 


77S114 
780222 


35-14 


221886 


34 


27 


779434 


34 


84 


999212 


• i3 


34-97 
34- 80 


219778 


33 


28 


781524 


34 


.67 


999205 


• i3 


782320 


217680 


32 


29 


7836o5 


34 


5i 


999197 


•i3 


784408 


34-64 


215592 


3i 


3o 


7 856 7 5 


34 


3i 


999189 


• i3 


786486 


34-47 


2i35i4 


3o 


3i 


3- -787736 


34 


18 


9-999181 


• i3 


8- 7 88554 


34-3i 


1 1 «2i 1446 


29 


32 


789787 

791828 


34 


02 


999174 


•i3 


790613 


34-i5 


20938T 


28 


33 


33 


86 


999 1 66 


-i3 


792662 


33-99 
33-83 


207338 


27 


34 


79385 9 


33 


70 


999158 


• i3 


794701 


205299 


26 


35 


7 9 588i 


33 


54 


999 1 5o 


• i3 


796731 


33-68 


203269J 25 


36 


797894 


•33 


39 


999142 


• i3 


798752 
800763 


33-52 


201248 


24 


ll 


799*97 


33 


23 


999i34 


• i3 


33-3 7 


199237 


23 


801892 


33 


08 


999126 


-13 


802765 


33-22 


197235 


22 


39 


8o38 7 6 


32 


93 


Q99 1 '8 


• i3 


8o4758 


33-07 


195242 


21 


4o 


8oo852 


32 


78 


9991 10 


• i3 


806742 


32-92 


193208 20 


4i 


8-807819 


32 


63 


9-999102 


• i3 


8-808717 


32-78 


11 • 191 283 J 19 
189317 18 


42 


809777 


32 


49 


999094 


•14 


8io683 


32-62 


43 


811726 


32 


34 


999086 


•14 


81 2641 


32-48 


187359 17 


44 


8 1 366 7 


32 


10 


999077 


• 14 


8i458g 


32-33 


i854n! 16 


45 


815099 


32 


o5 


999069 


•14 


816529 


32-o5 


1 8347 1 | i5 


46 


817522 


3i 


91 


999061 


•14 


8 1 846 1 


181 539j 14 


3 


8i 9 436 


3i 


77 


999053 


•14 


820384 


3i- 9 i 


179616 i3 


821343 


3i 


63 


999044 


•14 


822298 


3i -77 


177702 1 12 


49 


823240 


3i 


4q 


999030 


•14 


824205 


3i-63 


175795; 11 
173897 10 


5o 


825i3o 


3i 


35 


999027 


•14 


826103 


3i-5o 


5i 


8-827011 


3i 


22 


9-999019 


•14 


8-827992 


3i -36 


11 -1720081 
170126! 8 


52 


828884 


3i 


08 


999010 


•14 


829874 


3i-23 


53 


830749 


3o 


f, 


999002 


•14 


83i-48 


3i-io 


168252 7 


54 


832607 


3o 


998993 


•14 


8336 1 3 


3o-o6 


166387 6 


55 


834456 


3o 


69 


998984 


•14 


835471 


3o-S3 


164529 5 


56 


836297 
838i3o 


3o 


56 


998976 
998967 


•14 


837321 


30-70 


162679! 4 


u 


3o 


43 


•i5 


83 9 i63 


3o-57 


160837 3 


83 99 56 


3o 


3o 


998908 


• i5 


840908 


3o-45 


l5q002| 2 


59 


841774 


3o 


17 


998950 


-i5 


842825 


3o-32 


1 57 1 75J I 


6o 


843585 


3o 


00 


99 8 94i 

Sine 


• i5 


844644 


80-19 


i55356| 


Cosino 


_D 


86° 


QolMlg. _ 


Tang. 1 


M. _, 



2 


(i , 


3EGREE6 


.) A TABLE OF LOGARITHM) 






M. 




Sine | 
8-843585 


D. i 

3o-o5 


Cosine 1 
9-998941! 


D. Tang. 


3o- 19 


Co tang. 1 




•i5i 8-844644 


ii-i55356 60 


i 


845387 


29-92 


998932! 


• i5 846455 


30-07 


153545, 5o 
1 51740! 58 


2 


847183 


29-80 


99 8 923, 


-i5 848260 


29-95 


3 


848971 


20-67 


998914: 


•i5 85oo57 


29-82 


1 4994 3 l 5 7 


4 


85oi5i 


29-55 


998905 
998896 


•i5 85i846 


29.70 


1 481 54 56 


5 


85a 525 


29-43 


•i5 853628 


29-58 


146372 55 


6 


834291 


29-31 


998887 


.i5| 8554o3 


29.46 


1 443971 54 


7 


856049 


29-19 


998878 


•i5 837171 


29-35 


142819 s 53 
141008, 52 


8 


867801 


29-07 
28-96 


998869 


-i5 858 9 32 


29-23 


9 


85 9 546 


99S860 


-i5 860686 


29-11 


139314' 5i 


10 


86i283 


28-84 


998801 


•i5 862433 


29 -cc 


1 37567 : 5o 


1 1 


8-863oi4 


2 8- 7 3 


9-998841 


• i5 8-864n3 


28-88 


11 135827! 49 
1 34094 1 48 


12 


864738 


28-61 


998832 


•15, 865906 


28.77 


i3 


866455 


28 -5o 


998823 


.16 86 7 632 


28-66 


132368 47 


M 


868 1 65 


28-39 
28-28 


998813 


•16 869351 


28.54 


130649 


46 


i5 


869868 


998804 


•16 871064 


28-43 


128936 


45 


16 


871665 


28-17 


998795 


■16! 872770 


28-32 


127230 


44 


»2 


873255 


28-06 


998785 


.16 874469 


28-21 


I2553i 


43 


18 


8 7 4 9 38 


27-95 


998776 


■16 1 876162 


28-11 


123838 


42 


'9 


876610 


27-86 


998766 


•16 877849 


28-00 


I22l5l 


41 


2t> 


878285 


2 7 - 7 3 


998757 


•16 879529 


27-89 


I 2047 1 


40 


21 


8-879949 


27-63 


9-99 8 747 


.16 8-881202 


27.79 
27.68 


II I 18798 


3 9 


22 


881607 


27-52 


998738 


•i6i 882869 


117131 


38 


23 


883258 


27-42 


998728 


■ i6j 88453o 


27.58 


116470 


37 


24 


884903 
886542 


27 -3i 


998718 


•i6| 886i85 


27-47 


ii38i5 


36 


25 


27-21 


998708 


•16 88 7 833 


27.37 


1 12167 


35 


26 


888174 


2711 


998699 


•16' 889476 


27-27 


no524 


34 


^ 


889801 


27-00 


998689 


•16 891112 


27.17 


108888 


33 


891421 


26-00 
26-80 


998679 


•i6| 892742 


27-07 


107268 


32 


29 


8 9 3o35 


998669 


•17, 8 9 4366 


26-97 


io5634 


3i 


3o 


894643 


26-70 


998659 


-17 8 9 5 9 84 


26-87 


104016 


3o 


3i 


8-896246 


26-60 


9-998649 


•17; 8-897696 


26-77 


11 -102404 


29 


32 


807842 


26-5i 


998639 


•171 899203 


26-67 


100797 


28 


33 


899432 


26-41 


998629 


•17 900803 


26-58 


099197 


27 


34 


901017 


26-3i 


9986 1 9 


- 1 -7 1 902398 

•17J 903987 
•17 905370 


26-48 


097602 


26 


35 


902696 


26-22 


998609 


26-38 


096013 


25 


36 


904169 


26-12 


998399 


26-29 


094430 


24 


37 


905736 


26 -o3 


998589 
998078 


•17I 907147 


26-20 


092853 


23 


38 


907297 


25-o3 


.17 908719 


26-10 


091281 


22 


3 9 


908853 


25-84 


99 8568 


•17 910285 


26-01 


089715 


21 


4o 


910404 


25-75 


998558 


•17 911846 


25 -92 


088 1 54 


20 


4i 


8-911949 
913488 


25-66 


9-998548 


- 1 -7 j 8-913401 


25-83 


1 1 -086599 


\l 


42 


25-56 


998537 


-171 914951 


25-74 


o85o4o 
o835o5 


43 


916022 


25-47 


998527 


•17 916495 


25-65 


17 


44 


9i655o 


25-38 


99 85i6 


•18! 918034 


25-56 


081966 


16 


45 


918073 


25-29 


998506 


•18 9 i 9 568 


25-47 
25-38 


080432 


i5 


46 


919391 


25-20 


998405 


•18 921096 


078904 


14 


47 


921 io3 


25-12 


998485 


•18 922619 


25-3o 


077381 


i3 


48 


922610 


25-o3 


998474 


•18 924136 


25-21 


075864 


12 


49 


924112 


24-04 


998464 


•18 926649 


25-12 


074351 


11 


5o 


920609 


24-86 


998453 


•18 927156 


25-o3 


072844 


10 


5i 


8-927100 


24-77 


9-998442 


•18' 8-928658 


24-95 
24-86 


n -071342 


I 


52 


9 2858 7 
93oo63 


24-69 


998431 


•18 93oi55 


069845 
o68353 


53 


24-60 


998421 


-i8| 9 3i647 


24-78 


I 


54 


93 1 544 


24-52 


998410 


.181 9 33i34 


24-70 


066866 


55 


933oi5 


24-43 


998399 
998388 


•18 934616 


24-61 


065384 


5 


56 


934481 


24-35 


.18! 936093 


24-53 


06390-7 


4 


ii 


935942 


24-27 


998377 


.18! 9 3 7 565 


24-45 


o62435 


3 


937398 
9 3885o 


24-19 


998366 


•18 939032 


24-37 


060068 


2 


59 


24-u 


998355 


•181 940494 
•18I 941932 

85°, Cotansr. 


24 -3o 


069306 i 1 
1 o58o48' 


6o 


940296 


24 -o3 
D. 


998344 


24-21 

D. 


_.. 


■ Cosine 


Sine 


L T an £- 


: M. 





SINES 


AiND TANGENTS. 


(5 DEGREES.) 




23 


u~ 


Sine 


D. 


Cosine 


D. | 


Tang. 


D. 


Cotang. 


j 





8 940296 
941738 


24 


o3 


9-998344 


•19! 


8-941952 


24-21 


11 '038048' 60 | 


I 


23 


% 


998333 




l 9\ 


943404 


24 


i3 


056596 


3 


2 


Q43i74 


23 


998322 




19 


944852 


24 


o5 


o55i48 


3 


944606 


23 


79 


998311 




19. 


946295 


23 


97 


©537o5 


57 


4 


946034 


23 


7* 


998300 




l 9 


947734 


23 


82 


o52266 


56 


5 


947456 


23 


63 


998289 




19 


949168 


23 


©5o832 


55 


6 


948874 


23 


55 


998277 




'9 


950597 


23 


74 


049403 


54 


7 


95o2«7 


23 


48 


998266 




19 


952021 


23 


66 


047979 
046559 


53 


8 


95169^ 


23 


40 


998255 




19 


953441 


23 


60 


52 


9 


953ioo 


23 


32 


99 82 43| 


19 


954856 


23 


5i 


045 1 44 


5i 


10 


9 54499 


23 


25 


998232! 


19 


936267 


23 


44 


043733 


5o 


ii 


8 .955894 
957284 


23 


'7 


9.998220 


'9 


8-957674 


23 


37 


11-042326 


% 


12 


23 


10 


998209 


IQ 


939075 


23 


29 


040925 
039327 


i3 


958670 


23 


02 


998 1 971 
998186 


19 


960473 


23 


23 


47 


14 


960052 


22 


9^ 


19 


961S66 


23 


14 


o38i34 


46 


id 


961429 


22 


88 


998174' 


19 


963255 


23 


07 


036745 


45 


16 


962801 


22 


80 


99 8i63 


19 


964639 


23 


00 


o3536i 


44 


17 


96417c 


22 


73 


998101 ' 


'9 


966019 


22 


9 3 


033981 


43 


18 


965534 


22 


66 


998139 


20 


9 6 7 3 9 4 
968766 


22 


86 


o326o6 


42 


19 


966893 


22 


5 9 


998128 


20 


22 


79 


o3i234 


41 


20 


968249 


22 


52 


9981 16 


20 


97oi33 


22 


7i 


029867 
il-0285o4 


40 


2l 


8-969600 


22 


44 


9-998104, 


20 


8 -.071406 


22 


65 


18 


22 


970947 


22 


38 


$XI 


20 


972835 


22 


i 1 


027145 


23 


972289 
973628 


22 


3i 


20 


974209 


22 


5i 


025791 


37 


24 


22 


24 


99806S; 


20 


97556o 


22 


44 


024440 


36 


25 


974962 


22 


'7 


998056 


20 


976906 


22 


37 


023094 


35 


26 


976293 


22 


10 


998044! 


20 


978248 


22 


3o 


021762 


34 


2 7 


977619 


22 


o3 


998032 


20 


979586 


22 


23 


020414 


33 


28 


978941 


21 


97 


998020 


20 


980921 


22 


17 


019079 


32 


o9 


980259 
q8i5 7 3 


21 


90 


998008 


20 


982251 


22 


10 


017749 


3i 


3o 


21 


83 


997996, 
9. 997985 j 


20 


983577 


22 


04 


016423 


3o 


3i 


8-982883 


21 


77 


20 


8-984899 


21 


97 


1 1 -013101 


3 


3 2 


984189 


21 


70 


997972, 


20 


986217 


21 


91 


013783 


33 


985491 


21 


63 


9979 5 9I 


20 


987532 


21 


84 


OI2468 


2- 


34 


986789 


21 


57 


997947 


20 


988842 


21 


78 


i i i 58 


26 


3d 


9 88oS3 


21 


5o 


997935 


21 


990149 


21 


7i 


ooo85i 
008549 


25 


36 


989374 


21 


44 


997922 


21 


99i45i 


21 


65 


24 


ll 


990660 


21 


38 


997910 


21 


992750 


21 


58 


007250 


23 


38 


991943 


21 


3i 


997897 


21 


994045 


21 


52 


oo5955 


22 


3 9 


993222 


21 


25 


997885, 


21 


993337 


21 


46 


oo4663 


2! 


4o 


994497 


21 


19 


997872 


21 


996624 


21 


40 


003376 


20 


4i 


8-995768 


21 


•12 


9-997860 


21 


8-997908 


21 


34 


11 -002092 


\l 


42 


997036 


21 


.06 


997847: 


21 


999188 


21 


27 


000812 


43 


998299 


21 


00 


997835 


21 


9-000465 


21 


21 


20-999535 


17 


44 


999500 


20 


% 


997822 
997809, 


21 


001738 


21 


i5 


998262 


16 


45 


9-000816 


20 


21 


003007 


21 


09 


996993 


i5 


46 


002069 


20 


•82 


997797! 


21 


004272 


21 


o3 


995728 


14 


47 


oo33i8 


20 


76 


997784' 


21 


oo5534 


20 


97 


994466 


i3 


48 


oo4563 


20 


"O 


997771 ! 


21 


006792 


20 


9i 


993208 


12 


49 


oo58o5 


20 


6^ 


997758; 


21 


008047 


20 


85 


991953 


1 1 


5o 


007044 


20 


-58 


99774^: 


21 


009298 


20 


80 


990702 


10 


5i 


9-008278 


20 


52 


9-997732 1 


21 


9-010546 


20 


74 


10-989454 


I 


5s 


0095 1 


20 


•46 


9977J9 


21 


01 1790 


20 


68 


988210 


53 


010737 


20 


•40 


997706 


21 


oi3o3i 


20 


62 


986969 


7 


54 


011962 


20 


•34 


997693, 


22 


014268 


20 


56 


985732 


6 


55 


01 3 1 82 


20 


•29 


997680! 


22 


oi55o2 


20 


5i 


984498 


5 


56 


014400 


20 


•23 


997667 | 


22 


016732 


20 


45 


983268 


i 


57 


oi56i3 


20 


•n 


997634; 


22 


017959 


20 


40 


982041 


58 


016824 


20 


• 12 


997641 < 


22 


oi 9 i83 


20 


33 


980817 


2 


? 9 


oi8o3i 


20 


-06 


997628 


2 2 


020403 


20 


28 


979597 


1 


6c 


019235 


20 


•OO 


997614; 


22 


021620 


20 


23 


978380 







Cosine 


D. 


Sine [84°i 


Cotang. 


I). 


Tacsr. 


MiT 



24 


(G 


DEGREES.) A TABLE 


OF LOGARITHMIC 




M. 


Sine 


J). 


Cosine 


D. 


Tang. 


I). 


Cotang. 







9-019235 


20-00 


9-997°i4 


• 22 


9-021620 


20-23 


10-978380 


~6(T 


i 


020435 


19 


9 5 


997601 


• 22 


0*2834 


20 


•n 


977166 


& 


3 


02I632 


r 9 


89 


997588 


•22 


024044 


20 


1 1 


975g56 


3 


022825 


l 9 


84 


997574 


• 22 


02525l 


20 


06 


974740 
973545 


57 


4 


024016 


*9 


78 


997561 


• 22 


. 026455 


20 


00 


56 


5 


025203 


19 


73 


997547 


• 22 


02i655 


>9 


95 


972345 


55 


6 


026386 


l 9 


67 


997534 


•23 


028852 


'9 


90 


971 148 14 


7 


027567 


J 9 


62 


997520 


-23 


o3oo46 


•9 


85 


969954' 53 


8 


028744 


»9 


5 7 


9975o7 


•23 


o3i237 


19 


79 


968763 j 52 


9 


029918 


'9 


5i 


997493 
997480 


-23 


o32425 


>9 


74 


967575 


Si 


10 


031089 


19 


47 


-23 


033609 


'9 


69 


966391 


5o 


ii 


9>o32257 


>9 


41 


9-997466 


•23 


9-034791 


] 9 


64 


10-965209 


40 


12 


o3342i 


«9 


36 


9974^2 


-23 


033969 


*9 


58 


96403 1 


48 


i3 


o34582 


!9 


3o 


997439 


-23 


037144 


'9 


53 


962856 


47 


14 


035741 


J 9 


25 


997425 


-23 


o383i6 


'9 


48 


96 i 684 


46 


i5 


036896 


>9 


20 


99741 1 


-23 


o3 9 485 


'9 


43 


9605 1 5 


45 


16 


o38o48 


*9 


i5 


997397 


•23 


o4o65 1 


'9 


38 


9 r )n349 
958187 


44 


17 


039197 


'9 


10 


997383 


•23 


041813 


'9 


33 


43 


iS 


040342 


;g 


o5 


997369 


•23 


042973 


19 


28 


937027 


42 


•9 


041485 


99 


99735:) 


•23 


044 1 3o 


'9 


23 


935870 


41 


20 


042625 


18 


9 4 


997341 


•23 


045284 


•9 


18 


954716 


40 


21 


9-043762 


18 


9-997327 


•24 


9 • 046434 


'9 


i3 


10-953566: 39 


22 


044895 


18 


84 


9973 13 


• 24 


047582 


•9 


08 


9 524i8; 38 


23 


046026 


18 


79 


697299 


•24 


048727 


•9 


o3 


951273 37 


24 


047 ' 54 


.8 


73 


997283 


•24 


049869 
o5ioo8 


18 


98 


95oi3i 36 


25 


048279 


18 


70 


997271 


•24 


18 


$ 


948992; 35 
9 47856| 34 


26 


049400 


18 


65 


997257 


•24 


o52i44 


18 


3 


050D19 


18 


60 


997242 


•24 


053277 


18 


84 


946723, 33 


o5i63d 


18 


55 


997228 


•24 


054407 


18 


79 


945593; 32 


29 


052749 


18 


5o 


997214 


•24 


o55535 


18 


74 


9444651 3 1 


3o 


o53859 


18 


45 


997199 


•24 


o56659 


18 


70 


94334i! 3o 


3i 


9-004966 


18 


4i 


9-997183 


•24 


9.057781 


18 


65 


10-942219 29 


32 


056071 


18 


36 


997170 


•24 


058900 


18 


n 


941100J 28 


33 


007172 


18 


3i 


997 1 56 


•24 


060016 


18 


939984I 27 
938870J 26 


34 


058271 


18 


27 


997141 


•24 


061 i3o 


18 


5i 


35 


059367 


18 


22 


997127 


•24 


062240 


18 


46 


937760I 25 


36 


060460 


18 


17 


997112 


•24 


o63348 


18 


42 


936652 j ?4 


37 


o6i55i 


18 


i3 


997098! 


•24 


o64453 


18 


37 


935547 


23 


38 


062639 


18 


08 


997083 ; 


•25 


o65556 


18 


33 


934444 


22 


3 9 


063724 


18 


04 


997068 : 


•25 


o66655 


18 


28 


933345 


21 


4o 


064806 


17 


99 


997053! 


•25 


067752 


18 


24 


932248 


20 


4i 


9-065385 


17 


o4 


9-997039! 


•25 


9-068846 


18 


«9 


io-93ii54 


19 


42 


06696; 


17 


00 


997024 


•25 


069938 


18 


i5 


930062 


18 


43 


o68o36 


1/ 


S6 


(-,.,7009 


•25 


071027 


18 


10 


928973 
927887 


17 


44 


069 1 07 


•7 


81 


';o6994: 


•25 


0721 1 3 


18 


06 


16 


45 


070176 


17 


77 


9o6«)79 


•25 


073197 


18 


02 


926803 


i5 


46 


071242 


»7 


72 


996964, 


•25 


074278 


'7 


97 


925722 


14 


47 


072306 


'7 


68 


996049, 


•25 


o 7 5356 


17 


o3 


924644 


1 3 


48 


073366 


17 


63 


996934 


•25 


076432 


17 


89 


923568 


12 


P 


074424 


17 


5 9 


996919 


•25 


077505 


•7 


84 


922495 


11 


5o 


075480 


'7 


53 


996904! 
9.096889! 


•25 


078576 


17 


80 


921424 


10 


5i 


9-076533 


•7 


5o 


•25 


9-079644 


!7 


76 


10-920356 


* 


52 


077583 
078631 


[T 


46 


096874 ' 


•25 


080710 


n 


72 


919290 


53 


H 


42 


996838! 


•23 


081773 


H 


3 


918227 


I 


54 


079676 


17 


38 


996843' 


•25 


oS2833 


H 


9.1 7 1 67 


55 


080719 


'7 


33 


996828 


•25 


083891 


H 


59 


916109 


5 


56 


081739 


17 


20 
23 


9968 1 2 


.26 


084947 


17 


55 


9i5o53 


4 


57 


082797 
o83832 


17 


996797 


.26 


086000 


•7 


5i 


914000 


3 


58 


17 


21 


996782 


26 


087050 


17 


47 


9i2g5o 


2 


5 9 


084864 


H 


17 


996766, 


• 26 


088098 


H 


43 


911902' 1 


6o 


085894 


H 


.3 


996751 j '26 


089144 


17 


38 


9io856| 




Cosine 


D 


. 


Sine 8 3° 


Cotang. j 


D. 


Tang. J 


M. 



BINES AND TANGENTS. (7 DEUREES.j 



25 



M. 


Sine 


D. 


Cosine 


D. 


Tang. 


1 I>- 


Cotang. | 





9-080894 


i 7 -i3 


9-996751 


•26 


1 9-089144 


17-38 


10-9108561 60 


i 


086922 


17-09 


996735 


• 26 


090187 


17-34 


90081 3! 5q 


2 


087947 


17-04 


996720 


.26 


091228 


i7-3o 


908772 58 


3 


088970 


17-00 


996704 


-26 


092266 


17-27 


907734 57 


4 


089990 


16-96 


996688 


•26 


093302 


17-22 


906698 5o 


5 


091008 


i6-q2 


9Q6673 


•26 


! 094336 


17-19 


905664' 55 


6 


092024 


16-88 


996607 


•26 


095367 


17-13 


904633! 34 


3 


093037 


16-84 


99664 1 


•26 


096395 


17-11 


90360OI 53 


094047 


16-80 


996625 


-26 


097422 


17.07 


902578J 52 


9 


095o56 


16-76 


996610 


•26 


098446 


17-03 


901 554! 5i 


!0 


096062 


16-73 


996094 


•26 


099468 


16-99 
i6- 9 5 


ooo532 5o 

io-8go5i3 49 

898496 48 


!1 


9*097065 


16-68 


9 -996578 


•27 


9-100487 


12 


098066 


16. 65 


996562 


•27 


ioi5o4 


16-91 


13 


099065 


16-61 


996546 


•27 


102519 


16-87 


897481, 47 


M 


100062 


16.57 


996030 


•27 


io3532 


16-84 


896468 46 


i5 


ioio56 


16.53 


9965 1 4 


•27 


104542 


16-80 


895458, 45 


16 


102048 


16.49 


996498 


•27 


io555o 


16-76 


894450 


44 


>7 


io3o37 


16-45 


996482 


•27 


io6556 


16-72 


893444 


43 


18 


104025 


».6-4i 


996465 


•27 


107559 


16-69 


892441 


42 


•9 


io5oio 


16-38 


996449 


•27 


io856o 


i6-65 


891440 


4i 


20 


io5qq2 


i6-34 


996433 


•27 


109559 


1661 


890441 
10-889444 


40 


21 


9- 106973 


i6-3o 


9-996417 


•27 


9- 1 ioo56 


i6-58 


39 


22 


107951 


16-27 


996400 


•27 


1 1 1 55 1 


16.54 


888449 38 


23 


10S927 


16-23 


996384 


•27 


U2543 


i6- 5o 


887457 


37 


24 


109901 


16-19 


99 6368 


•27 


n3533 


16-46 


886467 


36 


25 


1 10873 


16-16 


99635i 


•27 


U452I 


i6-43 


835479 
884493 


35 


26 


1 1 1842 


16- 12 


996335 


•27 


1 i55o7 


16-39 


34 


27 


1 1 2809 


16-08 


99 63 1 8 


•27 


1 1 649 1 


16. 36 


8835o 9 


33 


28 


1 13774 


i6-o5 


090 3o2 


• 28 


1 17472 


16-32 


882528 


32 


29 


.i4 7 3 7 


16-01 


996285 


-28 


11S452 


16-29 
16-25 


881548 


3i 


3o 


1 1 56 9 8 


I5-Q7 


996269 


.28 


1 19429 


88o5 7 i 


3o 


3i 


9- n 6656 


15.94 


9-996252 


.28 


9-120404 


16-22 


10-879596 


3 


32 


iit6i3 


15.90 


996235 


•23 


121377I 


l6-l8 


878623 


33 


118067 


i5-87 


996219 


.28 


122343 


i6-i5 


8 77 652 
8 7 6683 


27 


34 


119519 


1 5 • 83 


996202 


•28 


123317. 


16- 1 1 


26 


35 


1 20469 


i5-8o 


99&i85 


-23 


124284 


16-07 


875716 


25 


36 


121417 


l5- 7 6 


996168 


• 28 


125249: 


16-04 


874751 


24 


ll 


122362 


1 5 - 73 


996151 


.28 


126211! 


16-01 


873780 
872828 


23 


i233o6 


1 5-69 


996134 


.28 


127172 


15-97 


22 


39 


124248 


i5-66 


9961 17 


.28 


i28i3o 


1 5 -94 


871870 


21 


4o 


125187 


i5-62 


9961 00 


'.28 


129087; 


15.91 

10-87 


870913 


20 


4i 


9-126125 


i5- 39 


9-996083 


• 29 


9-i3oo4ii 


10-869959 


19 


42 


127060 


1 5 - 56 


996066 


•29 


i3o994| 


15-84 


869006 


18 


43 


127993 


l5- 32 


996049 


•29 


i3i944| 
i328g3j 


i5-8i 


868o56 


17 


44 


128925 


I5-4Q 


996032 


•29 

•29 


15.77 


867107 


16 


45 


I2 9 §54 


15-45 


996015 


i3383 9 | 


15-74 


866161 


:5 


46 


1.^0781 


i5-42 


993908 


• 29 


i34784| 


1 5 • 7 1 


8652i6 


U 


47 


131706 


i5-3 9 


995980 


■29 


135726 


l5.6 7 


864274 


i3 


43 


i3263o 


15-35 


995963 


• 29 


1 3666 7 


15-64 


863333 


12 


49 


i3355i 


i5-32 


995946 


.29 


i376o5i 


i5-6i 


862395 
861458 


11 


5o 


134470 


15.29 


995928 


• 29 


138542 


15-58 


10 


5i 


9-135387 
i363o3 


l5-20 


9-995911 
9 9 58 9 4 


.29 


9-139476 


i555 


io-86o524 


I 


52 


15-22 


.29 


1 40409 | 


i5-5i 


809591 


53 


i3t2i6 


i5- 19 


995876 


•29) 


i4i34o' 


i5-48 


85866o 


7 


54 


i38i28 


i5-i6 


995859 


•29! 


142269 1 


i5-45 


857731 


6 


55 


139037 


I5-I2 


995841 


•29! 


143196! 


i5-42 


8568o4 


5 


56 


139944 


13-09 


995823 


.29 


i44i2i| 


1 5 -39 


805879 


4 


u 


Uo85o 


i5-o6 


995806 


•29 


i45o44 


i5-35 


854956 


3 


141 754 


i5-o3 


9 9 5 7 88 


• 29 


145966 
146385, 


i5-32 


854034 


2 


59 


142655 


i5-oo 


995771 


•20 


10-29 


853u5j 1 


oo 


143555 


14-96 


995753 ' .29 


147803 


l5-20 


852197; 




Cosine 


D. 


Sine !82°| 


Cotung. 1 


p: 1 


_Tang. _ 


JL 



20 


(8 


DEGKEUS.) A TABLE OF LOGARITHMIC 







Sine 
o- 143555 


D. 


Cosine 


D. | Tang. 


D. 


Cctang. 




14-96 


9-995753 


•3o o- 


147803 


i5-26 


I0'852io7 
85i28a 


00 


i 


144453 


14 


9 3 


995735 


-3o 


148718 


i5 


23 


1 


2 


145349 


14 


90 


995717 


•3oj 


1 4963 2 


i5 


20 


85o368 


3 


146243 


14 


87 


995699 
995681 


■ 3o 


i5o544 


i5 


17 


849456 


n 


4 


I47I36 


14 


84 


•3o 


i5i454 


i5 


14 


848546 


5 


148026 


14 


81 


995664 


• 3o 


1 52363 


i5 


1 1 


847637 


55 


6 


148915 


14 


78 


995646 


■3o 


153269 


i5 


08 


846 7 3i 


54 


J 


149802 


14 


75 


995628 


-3c 


1 54 1 74 


i5 


o5 


845826 


53 


1 30686 


14 


72 


99 56 10 


•3o 


i55o77 


i5 


02 


844923 


5j 


9 


i5i569 


14 


69 


995591 


-3o 


155978 


14 


99 


844022 


5i 


IO 


i5245i 


14 


66 


99.5573 


• 3o 


1 568 77 


14 


96 


843i23 


5o 


ii 


j i5333o 


14 


63 


9-995555 
995537 


• 3o o- 


l5 777 3 


14 


9 3 


10-842225 


48 


12 


1 54208 


14 


60 


-3o 


1 5867 1 


14 


90 


841329 


i3 


i55o83 


14 


5 7 


993319 


• 3o 


1 59565 


14 


87 


840435 


47 


U 


155957 


14 


54 


Q955oi 


.3i| 


160457 


• 4 


84 


83 9 543 


46 


i5 


1 5683o 


14 


5i 


993482 


• 3i 


i6i347 


14 


81 


838653 


45 


16 


157700 
1 5856o 


14 


48 


995464 


• 3i 


162236 


14 


79 


837764 


44 


\l 


14 


45 


993446 


• 3i' 


i63i23 


14 


76 


836877 


43 


i5g43D 


14 


42 


995427 


•3i ! 


164008 


14 


73 


835992 


42 


'9 


i6o3oi 


14 


3 9 


995409 


-3i 


164892 


14 


70 


83 5 1 08 


41 


20 


161164 


14 


36 


993390 


-3i 


i65 77 4 


14 


67 


834226 


40 


21 


9- 162025 


14 


33 


9-993372 


• 3i 9- 


166634 


• 4 


64 


10-833346 


U 


2 2 


162885 


14 


3o 


995355 


■ 3i; 


167532 


14 


61 


832468 


-.3 


163743 


14 


27 


995334 


•3i 


168409 


U 


58 


83 1 5 9 i 


ll 


24 


164600 


14 


24 


9953i6 


• 3i 


169284 


14 


55 


830716 


2 5 


165454 


14 


22 


995297 


• 3i| 


170157 


14 


53 


829843 
828971 


35 


26 


1 663o 7 


14 


l 9 


993278 


• 3i| 


17 1029 


14 


30 


34 


2 I 


167 1 59 


14 


16 


995260 


•3i| 


171899 


14 


47 


828101 


33 


28 


168008 


14 


i3 


995241 


.32! 


172767 


1 4 


4i 


827233 


32 


^9 


168856 


14 


10 


993222 


•32 


i 7 3634 


14 


42 


826366 


3i 


| 3o 


169702 


14 


07 


993203 


-32 


174499 


14 


3 9 


8255oi 3o 


1 3i 


9-170547 


14 


o5 


9-995184 


•32 ! 9. 


175362 


14 


36 


10-824638 29 
823776 28 


' 32 


I 7 i38 9 


14 


02 


995i65 


-.32 


176224 


14 


33 


I 33 


172230 


1 3 


99 


995146 


.32] 


177084 


14 


3i 


822916! 27 


34 


173070 


i3 


96 


993127 


•32| 


117942 


14 


2S 


822o58l 26 


1 35 


173908 


i3 


94 


995108 


-32 


'78799 
179653 


14 


25 


821201 


25 


36 


174744 


i3 


91 


995089 


•32 


14 


23 


820345 


24 


37 


i 7 55 7 8 


i3 


88 


993070 


-32 


180308 


14 


20 


819492 
818640 


23 


38 


17641 1 


i3 


86 


995o5i 


-32 


i8i36o 


14 


17 


22 


3 9 


177242 


i3 


83 


0g5o32 


-32 


18221 i 


14 


i5 


817789 


21 


40 


178072 


i3 


80 


9930i3| 


-32 


183039 


14 


12 


816941 


20 


4i 


9-178900 


i3 


77 


9- 99 499 3 


•32 9. 


183907 


14 


09 


10-81609.3 


;g 


42 


179726 
i8o55i 


13 


74 


99-4974 


-32 


184752 


14 


07 


8i5248 


43 


i3 


72 


994955 


.32 


185597 
186439 


14 


04 


8i44o3 


17 


44 


181374 


i3 


69 


994935 


• 3a! 


14 


02 


8i356i 


l6 


45 


[82196 


i3 


66 


9949 1 6 


•33; 


187280 


i3 


99 


812720! 13 


46 


i83oi6 


i3 


64 


99489 6 


•33 


188120 


i3 


96 


811880 14 


47 


183834 


i3 


61 


994877 


.33; 


i88 9 58 


i3 


93 


811042! i3 


48 


18465 1 


i3 


5 9 


994857 


• 33 


189794 


i3 


s 


810206J 12 


49 


185466 


i3 


56 


994838 


• 33 


190629 


1 3 


809.371 11 


5o 


18628c 


1 3 


• 53 


994818 


•31 


191462 


i3 


86 


8o8538j io 


5i 


9- 187092 


i3 


5i 


9.994798 


• 33 9* 


192294 


i3 


84 


10-^077061 9 
806876 1 8 


5a 


187903 


i3 


•48 


994779 
994739 


• 33 


193124 


1 3 


81 


53 


188712 


i3 


46 


• 33 


193953 


i3 


79 


806047! 7 


54 


189519 
I9c325 


.3 


• 43 


994789 


• 33 


194780 


i3 


76 


8o522o! 6 


55 


i3 


41 


994H9 


• 33 


195606 


•3 


74 


804394 ' 5 


56 


191130 


i3 


■38 


994700 


• 33 


196430 


i3 


7i 


803570, 4 


n 


1919.33 


i3 


• 36 


994680 


• 33 


197253 


i3 


69 


802747J 3 


192731 


i3 


• 33 


994660 


.33; 


198074 


i3 


66 


801926 2 


5 9 


■ 393534 


1 3 


•3o 


994640 


• 33! 


198894 


i3 


54 


801 106 


I 


6o 


ic.4332 


ii 


58 


994620 


• 33i 


i997i3 


i3-bi 


800287 


O 


1 Cosine 


E 


» 


Sine 1 


Bl°l C< 


;tang. 


D. 


Tang/ 


M 



SINES AND TANGENTS. (9 DEGREES.) 



27 



M, 


Sine 


D. 


Cosine 1 D. 


Tang. 


D. 


Ootang. 







g (94332 


i3- 


28 


9-994620 


33 


9-i997i3 


i3 


61 


•0-800287 


60 


l 


1961 ag 


i3- 


26 


994600 


33 


200529 


i3 


^ 


7994- 1 


5o 


2 


193920 


i3 


23 


99458o, 


33 


201340 


i3 


56 


7 9 8655 


58 


3 


196719 


i3- 


21 


994360 


34 


202159 


i3 


¥ 


797841 


57 


4 


197011 


i3- 


18 


994040 


34 


202971 


i3 


52 


797c 20 06. 

796*18, 00 


5 


198302 


i3 


'6 


994oi9 ! 


34 


203782 


i3 


49 


6 


1 9909 1 


1 j 


i3 


99 i499 


34 


204592 


i3 


47 


795408 54 


7 


199879 


i3 


11 


9944 79 


34 


2o54oo 


i3 


45 


794600 53 


8 


2oc666 


i3 


08 


994439 
994438 




34 


206207 


i3 


42 


793793 52 


9 


*>oi45i 


i3 


06 




34 


207013 


i3 


40 


792987 


31 


10 


202234 


i3 


04 


994418 




34 


207817 
9.208619 


i3 


38 


792183 


5o 


ii 


5 203017 


i3 


01 


9.994397 




34 


i3 


35 


io-79i38i 


% 


12 


203797 


12 


99 


994377 




34 


209420 


i3 


33 


790580 


i3 


204077 


12 


96 


994357 




34 


2IO?20 


i3 


3i 


789^80 
788982 


47 


14 


2o5354 


12 


94 


994336 




34 


2iioi8 


i3 


28 


46 


i5 


2o6i3i 


12 


87 


9943 1 6 




34 


2 : 1 8 1 5 


i3 


26 


788i85 


45 


16 


206906 
207679 


12 
12 


994295 
994274 
994254 




34 
35 


212611 
2i34o5 


i3 
i3 


24 
21 


787389 
786593 


44 
43 


208432 


12 


85 




35 


214198 


i3 


19 


785802 


42 


>9 


209222 


12 


82 


994233 




35 


214989 


i3 


17 


785011 


41 


20 


209992 


12 


80 


994212 




35 


215780 


i3 


i5 


784220 


40 


21 


9-210760 


12 


78 


9-994191 




35 


9-2i6568 


i3 


12 


10-783432 


IS 


22 


2Il526 


12 


7 5 


9941 7 1 




35 


217356 
218142 


i3 


10 


782644 


23 


212291 
2i3o55 


12 


73 


9941 5o 




35 


i3 


cS 


7 8i858 


37 


24 


12 


71 


994120 




35 


218926 


i3 


c5 


781074 


36 


25 


2i38i8 


12 


68 


994108 




35 


219710 


i3 


c3 


780290 


35 


26 


214579 
215338 


12 


66 


994087 




35 


2204Q2 


i3 


CI 


779308 
778728 


34 


2 7 


12 


64 


994066 




35 


221272 


12 


09 


33 


28 


216097 


12 


61 


994045 




35 


222o52 


12 


?7 


777948! 32 


I** 


216854 


12 


5 9 


994024 




35 


222830 


12 


?4 


777170 


3x 


3o 


217609 
9-2i8363 


12 


57 


994003 




35 


2236o6 


12 


?2 


776394 


3o 


3i 


12 


55 


9-993981 




35 


9.224382 


12 


90 


10-775618 


3 


32 


219116 


12 


53 


993960 




35 


225i56 


12 


88 


774844 


33 


219868 


12 


5o 


993939 
993918 




35 


225929 


12 86 


774071 


27 


34 


220618 


12 


48 




35 


226700 


12 84 


7733oo 


26 


35 


221367 


12 


46 


9 9 38 9 6 




36 


227471 


12 81 


772529 


25 


36 


222II5 


12 


44 


993875 




36 


228239 


1279 


771761 


24 


ll 


222861 


12 


42 


993854 




36 


229007 


12-77 


770993 


23 


2236o6 


12 


3 9 


9 9 3832 




36 


229773 

23oo3g 


11-76 


770227 


22 


3q 


224349 


12 


37 


99381 1 




36 


il- 7 3 


760461 

768698 

10-767935 


21 


4o 


225092 


12 


35 


993789 




36 


23l302 


12-71 


20 


4i 


9-225833 


12 


33 


9-993768 




3o 


9-232o65 


12-69 


\l 


42 


226573 


12 


3i 


993746 




36 


232826 


12.67 


767174 


43 


22731 1 


12 


28 


993723 




36 


233586 


ii-65 


766414 


17 


44 


228048 


12 


26 


993703 




36 


234345 


12-62 


765655 


l6 


45 


228784 


12 


24 


993681 




36 


235io3 


12-60 


764897 


i5 


46 


229018 


12 


22 


993660 




36 


235859 


.2-58 


764141 


i4 


% 


2302D2 


12 


20 


9 9 3638 




3o 


2366i4 


2-56 


763386 


i3 


230984 


12 


18 


993616 




36 


237368 


2-54 


762632 


12 


49 


23l 7 l4 


12 


16 


993594 




37 


238120 


12-52 


761880 


11 


5o 


232444 


12 


14 


993572 




37 


2388 7 2 


i2-5o 


761128 


10 


5i 


9-233172 


12 


12 


9-993500 




3^ 


9-239622 


12-48 


10-760378 


I 


52 


233899 


12 


09 


993528 




37 


240371 


12-46 


759629 


51 


234620 


12 


07 


9935o6 




37 


241 118 


12-44 


758882 


I 


54 


235349 


12 


o5 


993484 




37 


24i865 


12-42 


758i35 


55 


236073 


12 


o3 


903462 




37 


242610 


12-40 


757390 


5 


56 


236795 


12 


01 


993440 




37 


243354 


12-38 


756646 


4 


u 


237015 


ii 


99 


993418 




37 


244097 


12-36 


755903 


3 


238235 


11 


97 


993396 




37 


244839 


12-34 


755i6i 


2 


J9 


238953 


" 


9 5 


993374 




3^ 


245579 


12-32 


754421 


1 


6o 


239670 


11 


93 


99335 1 


•3 7 


246319 


12-30 


75368i 







Ocftine 


_P 




Sine 


80° 


Cotang. 


r 


>. 


_ Tang" 


M7j 



28 


(10 DEGREES.) A 


1ABLE OF LOGARITHMIC 


60 


M. 


Sine D 




Cosine 


D. 


Tang. 


D. 


Cotang. 


o 


9-239670 11 


9 3 


9»99335i 


I 7 


9- 240319 


12 -3o 


io-75368i 


i 


240386 1 1 


% 


993329 


•3-1 


247057 


12-28 


752943 5a 
752206! 58 


2 


241101 11 


993307 


•37 


247794 


12-26 


3 


241814 n 


87 


993285 


i 7 


24853o 


12-24 


75i47o 57 


4 


242526 11 


85 


993262 


I 7 


249264 


12-22 


7507.36 


56 


5 


243237 11 


83 


993240 


:ll 


24999 s 
25o73o 


12-20 


75ooo2 


55 


6 


243947 1 1 


81 


993217 
993195 


I2-I8 


740270 

748539 


54 


I 


244656 11 


79 


• 38 


25i46i 


12-17 


53 


245363 1 1 


]l 


993172 


-38 


262191 


I2-l5 


747809 


5a 


9 


246069 1 1 
246775 11 


993i49 


• 38 


252920 


1213 


747080 


5i 


10 


73 


99 3i2 7 


• 38 


233648 


12-11 


746352 5o 


II 


0-247478 11 
248181 11 


7i 


9-993104 


• 38 


9-254374 


12-09 


10-745626; 49 
744900 48 


12 


69 


993081 


• 38 


255ioo 


12-07 

12 o5 


i3 


248883 1 1 


ti 


993059 


•38 


255824 


744176 


% 


14 


249583 1 1 


993o36 


• 38 


256547 


12-03 


743453 


ID 


250282 11 


63 


9930 1 3 


• 38 


257269 


I2«0I 


742731 


45 


16 


250980 11 


61 


992990 


• 38 


257990 
258710 


12-00 


742010 


44 


17 


251677 ! 11 


it 


992967 


• 38 


II.98 


741290 


43 


18 


252373 11 


992044 


• 38 


259429 


II.96 


740571 


42 


l 9 


253067 11 


56 


992Q2I 


• 38 


260146 


11 -94 


73 9 854 


41 


20 


253761 n 


54 


992898 


• 38 


26o863 


11.92 


739137 
10-738422 


40 


21 


9-254453 11 


52 


9-992875 


• 38 


9-261578 


\\r 9 


ll 


22 


255i44 11 


5o 


992852 


• 38 


262292 


737708 


23 


255834 11 


48 


992829 


I 9 


263oo5 


U.87 


736995 
736283 


37 


24 


256523 11 


46 


992806 


19 


263717 


u-85 


36 


25 


25721 1 11 


44 


992783 


• 3 9 


264428 


n-83 


735572 


35 


26 


257898 11 


42 


992759 


I 9 


2 65 1 38 


n-8i 


734862 


34 


2 


258583 11 


4i 


992736 


• 3 9 


265847 


:::?? 


734i53 


33 


239268 n 


3 9 


992713 


i 9 


266555 


733445 


32 


29 


259951 11 


37 


992690 


•3 9 


267261 


n.76 


732739 
732o33 


3i 


3o 


26o633 1 1 


35 


992666 


•3 9 


267967 


u-74 


3o 


3i 


9-26i3i4 11 


33 


0-99^043 


•3 9 


9-268671 


11.72 


io-73i32o 
730623 


3 


32 


261994 11 


3i 


992619 


I 9 


269375 


11.70 


33 


262673 11 


3o 


992596 


•? 9 


270077 


11.69 


729923 


27 


34 


26335i n 


28 


992572 


•3 9 


270779 


:::!? 


729221 
728521 


26 


35 


264027 11 


26 


992549 

992523 


.39 


271479 
272178 


25 


36 


264703 11 


24 


I 9 


11.64 


727822 


24 


37 


265377 11 


22 


992501 


•3 9 


272876 


11-62 


727124 


23 


38 


266o5 1 11 


20 


992478 


.40 


273573 


1 1. 60 


726427 


22 


3 9 


266723 11 


«9 


992454 


.40 


274269 


u-58 


725731 


21 


40 


267395 11 


[I 


992430 


.40 


274964 


I!-55 


725o36 


20 


41 


9- 268o65 11 


9-992406 


•40 


9-275658 


10-724342 


\l 


42 


268734 11 


i3 


992382 


•40 


276351 


u-53 


723649 


43 


269402 1 1 


11 


992359 


.40 


277043 


11. 5i 


722957 


17 


4i 


270069 1 1 


10 


992333 


•40 


277734 


n-5o 


722266 


16 


45 


270733 11 


08 


99231 1 


•40 


278424 


11.48 


721576 


i5 


46 


271400 11 


06 


992287 


•40 


2791 i3 


w.% 


720887 


14 


% 


272064 11 


o5 


992263 


•40 


279801 
280488 


720199 


i3 


272726 11 


o3 


992239 


.40 


1 1-43 


71961 2 

718826 


12 


49 


2/3388 11 


01 


992214 


•40 


281174 


n-41 


11 


5o 


274049 10 


5 


992190 


.40 


28i858 


11-40 


718142 iO 


5i 


9 274708 10 


9-992166 


•40 


9-282542 


11. 38 


10 717458 


I 


52 


275367 10 


96 


992142 


•40 


283225 


n-36 


716775 


53 


276024 10 


94 


992117 


•41 


283907 
•)84588 


n-35 


716093 


7 


54 


276681 10 


92 


992093 


•41 


u-33 


7i54i3j 


55 


277337 10 


% 


992069 


•41 


285268 


ii-3i 


714732 


5 


56 


277991 10 
278644 10 


992044 


■41 


285947 
286624 


n-3o 


7 1 4o53 


4 


57 


87 


992020 


•41 


11.28 


713376 


3 


58 


279297 10 


86 


991996 


•41 


287301 


11.26 


712690 


2 


5 9 


279948 10 
280599 10 


84 


991971 


•41 


287977 


11-25 


71202J 


i 


60 


82 


991947 


•41 


288652 


11-23 


711348 





Cosine D 




Sine 


790 


Cotang. 


D. 


~Tang/ M. 



SIKE8 AND TANQEN T TS. (11 DEGREES.) 



29 



^7 


Sine 


i I). 


Cosine 


I). 


Tang. 


i D. 


Cot'ang. 


1 


o 


9 280D99 


10-82 


9-9Qigi7 


•41 


9-288652 


! 11-23 


.0-711348 60 


i 


281248 


10 


Si 


991922 


•41 


289326 


, 11-22 


710674: 5q 
7 1 0001 58 


2 


281897 


10 


79 


99189-7 


•4i 


i 289999 


1 I -20 


3 


282044 


10 


77 


991873 


•4» 


1 290671 


11-18 


709329 57 
7o8658, 56 


4 


283190 


10 


76 


991848 


•41 


, 291342 


n 17 


5 


283836 


10 


74 


991823 


•4i 


i 292013 


11. i5 


707987I 55 
707318 54 


6 


284480 


10 


72 


991799 


•4i 


292682 


11-14 


I 


285124 


10 


7i 


991774 


•42 


293350 


11-12 


7o665o 53 


280766 


10 


69 


991749 


•42 


294017 


11 -ii 


705983 


5a 


9 


286408 


10 


67 


991724 


•42 


294684 


11-09 


7o53i6 


5i 


10 


287048 


10 


66 


991699 


•42 


295349 
9-296013 


11-07 


70465i 


5o 


n 


5 287687 


10 


64 


9-991674 


•42 


n-o6 


10-703987 


% 


i? 


268326 


10 


63 


991649 


•42 


296677 


1 1 -o4 


7o3323 


i3 


288964 


10 


61 


991624 


•42 


297339 


n-o3 


702661 


47 


U 


289600 


10 


u 


991599 


•42 


' 298001 


II -01 


701999 
7oi338 


46 


i5 


290236 


10 


991574 


•42 


29S662 


11-00 


45 


16 


290870 


10 


56 


99 '549 


•42 


299322 


10.98 


700678 


44 


3 


29004 


10 


54 


99i524 


•42 


299980 


10-96 


700020 


43 


292137 


10 


53 


991498 
991473 


•42 


3oo638 


10-95 


699362 


42 


'9 


292768 


10 


5i 


•42 


301295 


10-93 


698705 


41 


20 


293399 


10 


5o 


99U48 


•42 


301961 


10-92 


698049 


4c 


21 


9-294029 
294658 


10 


41 


9-991422 


•42 


9-302607 


10-90 


10-697393 


3q 


22 


10 


46 


991397 


•42 


3o326i 


10-89 


696739 


38 


23 


295286 


10 


45 


991372 


•43 


3o3oi4 
3o4567 


10-87 


696086 


37 


24 


295913 


10 


43 


991346 


•43 


io«86 


695433 


36 


25 


296539 


10 


42 


99i32i 


•43 


3o52i8 


10-84 


694782 


35 


26 


297164 


10 


40 


991295 


•43 


3o586g 


io-83 


6941 3 1 


34 


27 


297788 


10 


39 


991270 


•43 


3o65i9 


io-8i 


693481 


33 


28 


298412 


10 


37 


991244 


•43 


307168 


10-80 


692832 


32 


2Q 


299034 


10 


36 


991218 


•43 


3o 7 8i5 


10.78 


692185 


3i 


3o 


299655 


10 


34 


991 193 


•43 


3o8463 


10-77 


691537 


3c 


3i 


9-300276 


10 


32 


9-991167 


•43 


9-309109 


10-75 


10-690891 


ll 


32 


300895 


10 


3i 


99"4i 


•43 


309754 


10-74 


690246 


33 


3oi5i4 


10 


29 


991 1 1 5 


•43 


310398 


10-73 


689602 


27 


34 


302132 


10 


28 


99 1 090 


•43 


3no42 


10-71 


688g58 


26 


35 


302748 


10 


26 


991064 


• 43 


3n685 


10-70 


6883 1 5 


25 


36 


3o3364 


10 


25 


99io38 


• 43 


3i2327 


10.68 


687673 


24 


ll 


3o3o79 
3o4393 


10 


23 


991012 


• 43 


312967 


10-67 


68 7 o33 


23 


10 


22 


990986 


•43 


3i36o8 


10-65 


6863 9 2 


22 


39 


3o5207 


10 


20 


990960 


•43 


3i4247 


10-64 


685753 


21 


40 


3o58i9 


10 


l 9 


990934 


•44 


3 1 4885 


10-62 


6S5 1 1 5 


20 


4i 


9-3o643o 


10 


5 7 


9-990908 


•44 


9-3i5523 


io-6i 


10-684477 


19 


42 


307041 


10 


16 


990882 


•44 


3 1 6 1 59 


1 • 60 


683841 


18 


43 


3o765o 


10 


14 


99oS55 


•44 


316793 


10-58 


683205 


H 


44 


308209 


10 


i3 


990829 


•44 


3 17430 


io-57 


682570 


16 


45 


308867 


10 


1 1 


99o8o3 


•44 


318064 


io-55 


681936 
68i3o3 


i5 


46 


309474 


10 


10 


990777 


•44 


318697 


io-54 


14 


s 


3 1 0080 


10 


08 


990750 


•44 


319329 


io-53 


680671 


i3 


3 io685 


10 


°7 


990724 


•44 


3 1 996 1 


io-5i 


680039 
679408 
678778 


12 


$ 


3:1289 


10 


o5 


990697 


•44 


320592 


io- 5o 


11 


5c 


3 1 1 8 9 3 


10 


04 


990671 


•44 


321222 


10-48 


10 


5i 


c-3i2 495 


10 


o3 


9-990644 


•44 


9-32i85i 


10-47 


10-678149 


9 


52 


3 1 3097 
313698 


10 


01 


990618 


•44 


322479 


io-45 


677521 6 


53 


10 


00 


990591 


•44 


323io6 


io-44 


676894I 7 


54 


314297 


9 


98 


990065 


•44 


323733 


10-43 


676267 


6 


55 


314897 


9 


97 


99o538 


•44 


324358 


10-41 


675642 


5 


50 


3 1 5495 


9 


06 


99o5u 


.45 


3249S3 


io-4& 


6 7 5oi 7 


4 


1 


816092 
3i668 9 


9 


94 


990485 


• 45 


325607 


10-39 


6 7 43 9 3 


3 


9 


9 3 


990458 


.45 


32623i 


io-37 


673769 


2 


59 


317284 


9 


9' 


990431 


•45 


326853 


io-36 


671147 
672526 


1 


60 


3i 7 8 79 


9 


90 


990404 


•45 


327475 
CojEansr. 


io-35 







Cosine 


D 




Sine 


7 8° 


' 1). 


Tang. 


IT" 



30 


(1* 


I DEGREES.) A 


TABLE OF LOGARITHMIC 




it 

o 


Sine 


D. 


| Cosine | IX 


| Tang. 


D. 


| Cotang 


60 


9-317879 


9.90 


9-99040*' -45' 9-32747^ 


io-35 


10-672526 


i 


3i8473 


9-88 


990378 -45, 328095 


10-33 


671905 


U 


2 


319066 


9-87 


99o35i -4f 


i 328 7 i5 


10-32 


671285 


3 


3i 9 658 


1 9-86 


99o324j -4' 


> 32 9 334 


io-3o 


670666 


n 


4 


320249 


9.84 


990207) -4f 


! 329953 


10-20 

10-28 


670047 


5 


320840 


9-83 


990270 -451 33o57o 


669430 
6688 1 3 


55 


6 


32i43o 


9-82 


990243 -4i 


33 1 1 87 


10.56 


54 


I 


322019 


9-80 


99021' 


.45 


1 33i8o3 


10-25 


668197 

667382 


53 


322607 


9-79 


990188 


•45 


3324i8 


10-24 


52 


9 


323194 

1 J23780 


9-77 


9901 61 


•45 


J 333o33 


10-23 


666967 
666354 


5i 


IC 


9-76 


990134 


•45 


333646 


10-21 


5o 


ii 


9-324366 


9-75 


9.990107 


■46 


! 9-334259 


10-20 


10-665741 


a 


i-i 


324950 
325534 


9-73 


990079 


.46 


334871 


io- 19 


665 1 29 
6645 1 8 


1 3 


9.72 


990052 


•46 


335482 


I0-I7 


47 


14 


326117 


9.70 


990025 
989997 


.46 


336093 


io- 16 


663907 


46 


i5 


326700 


9-69 


•46 


J36702 


.o-i5 


663298 
662689 


45 


16 


327281 


9.68 


989970 


•46 


33 7 3u 


io- 13 


44 


:i 


327862 


9-66 


989942 


•46 


337919 


10-12 


662081 


43 


328442 


9-65 


989915 


•46 


338327 


10- II 


661473 


42 


'9 


32902 j 


9.64 


989887 


•46 


339i33 


10- 10 


660867 


41 


20 


329399 


9-62 


989860 


•46 


339739 


10-08 


660261 


40 


?i 


9-330176 


9-61 


9-989832 


•46 


9- 34o344 


10-07 


10-659656 


& 


:2 


33ot53 


9-60 


989804 


•46 


340948 
34io52 


io- 06 


659052 


23 


33i32 9 
33i9o3 


9-58 


989777 


-46 


10-04 


658448 


37 


24 


9-5 7 


989749 


•47 


342i55 


10 -o3 


65 7 845 


36 


25 


332478 


9-56 


989721 


•47 


342757 


10-02 


657243 


35 


26 


333o5i 


9-54 


989693 


•47 


343358 


10-00 


656642 


34 


27 


333624 


9-53 


989665 


•47 


343g58 


9'99 


656o42 


33 


28 


334195 


9-52 


989637 


•47 


344558 


9.98 


655442 


32 


n 9 


334-766 


9-5o 


989609 


•47 


345i57 


9'97 


654843 


3 1 


3o 


33533- 


9-4Q 
9-48 


9 8 9 582 


•47 


345755 


9.96 


654245 


3o 


3i 


9-335906 


9-989553 


•47 


9-346353 


9-94 


10-653647 


3 


32 


336475 


9.46 


989525 


•47 


346949 
347043 


9-93 


653o5i 


33 


337043 


9-45 


989497 


•47 


9-92 


652455 


27 


34 


337610 


9.44 


989469 


•47 


34*141 


9-91 


65i85 9 


26 


35 


338176 


9-43 


989441 


•47 


348735 


?:g 


65 1 265 


25 


36 


33S742 


9-4i 


989413 


•47 


349329 


650671 


24 


37 


3393o6 


9.40 


989384 


•47 


349922 


9.87 


650078 


23 


38 


339871 


9-39 


989356 


•47 


35o5i4 


9-86 


649486 


22 


39 


340434 


9-37 


989328 


•47 


35i 106 


9-85 


648894 


21 


4o 


340996 


9-36 


989300 


•47 


351697 
9-352287 


9-83 


6483o3 


20 


4i 


9-341008 


9-35 


089271 


•47 


9.82 


10-647713 


\l 


42 


342119 


9-34 


' 989243 


•47 


352876 


9-81 


647124 


43 


342679 


9-32 


989214 


•47 


353465 


9-80 


646535 1 


44 


343239 


9-3i 


989186 


•47 


354053 


9-79 


645947 
64536o 


r6 


45 


343797 


9-3o 


9 8 9 i5 7 
989128 


•47 


354640 


9-77 
9.76 


i5 


46 


344355 


9-29 


•48 


355227 


644773 


14 


S 


344912 


9.27 


989100 


•48. 


3558i3 


9-75 


644187 


i3 


345469 


9-26 


989071 


-48 


3563o8 

356982 
357566 


9-74 


643602 


12 


49 


346024 


9-25 


989042 


•48 


9 . 7 3 


643ei8 :i 


5o 


346579 


9-24 


989014 


•48 


9-71 


6424341 10 


5i 


9 347134 


9-22 


9-988985 


•48 


9-358149 


9-70 


io-64i85i, 
641269! 8 


52 


347687 


9-21 


988956 


•48 


358 7 3i 


9-69 I 
9.68 


53 


348240 


9- 20 


988927 


•48 


3593i3 


640687 7 
640107 6 


54 


348792 


9.19 


988898 


-.48 


359893 


9.67 


55 J 


349343 


9.17 


988869 1 .48 


360474I 


9.66 


639526 5 


56 | 


349893 


9-16 


988840' .48 


36io53: 


9-65 


638947 1 4 


57 


35o443 


9- 15 


98881 1 .49 


36i632 


9-63 


638368 3 


58 ! 


350992 


9-14 


988782! .49 


362210 


9-62 


637790] 2 


5o 


35i 540 


9-i3 


988753 .49 1 


362787 


9-61 


6372i3| 1 


6o 


352088 


9. 11 


988724 -49 1 


363364: 


9.60 
Th j 


636636 1 




Cosine | 


D. 


Sine 17° 


Cotang^ 1 


Tang. 1, 


M. j 





SINES AND TANGENTS. 


(13 DEGREES. 


) 


3 


M. 




Sine 


__^_ 


Cosine 


D. 


Tang. 


D. 


Cotang. 




9-352088 


9. 11 


9-988724 


"^49 


9-363364 


9-60 


io-b36636 


"60" 


i 


352635 


9 


10 


988695 


.49 


363940 
3645 1 5 


9 


u 


636o6o 


% 


2 


353i8i 


9 


3 


988666 


•49 


9 


635485 


3 


353726 


9 


9 88636 


.49 


365ooo 


9 


57 


634910 


57 


4 


354271 


9 


07 


988607 


.49 


365664 


9 


55 


634336 


56 


5 


3548i5 


9 


o5 


988578 


•49 


366237 


9 


54 


633763 


55 


6 


355358 


9 


04 


9 88548 
988519 


•49 


3668 10 


9 


53 


633190 


54 


I 


355901 


9 


o3 


•49 


367382 


9 


52 


632618 


53 


356443 


9 


02 


988489 


•49 


367953 


9 


5i 


632047 


52 


9 


356o84 
357624 


I 


01 


988460 


•49 


368D24 


9 


5o 


631476 


5i 


IO 


99 


988430 


•49 


369094 


9 


4 2 
48 


630906 


5o 


ii 


9- 358o64 


8 


98 


9-988401 


•49 


9-369663 


n 


io-63o337 


% 


12 


3586o3 


8 


97 


988371 


•49 


370232 


9 


40 


629768 


i3 


359141 


8 


96 


988342 


•49 


370799 


9 


45 


629201 
628633 


47 


14 


359678 


8 


9 5 


9 883 1 2 


•5o 


371367 


9 


44 


46 


15 


36o2i5 


8 


93 


988282 


00 


3 7 i 9 33 


9 


43 


628067 


45 


16 


360702 


8 


92 


988252 


.50 


372499 


9 


42 


627501 


44 


\l 


361287 


8 


9i 


988223 


.50 


373o64 


9 


41 


626936 
626371 


43 


361822 


8 


1 


988193 


.50 


373629 


9 


40 


42 


*9 


362356 


8 


988163 


-50 


374193 


9 


u 


625807 


41 


20 


362889 


8 


9 88i33 


•5o 


3747^6 


9 


625244 


40 


21 


9-363422 


8 


87 


9 • 988 1 o3 


•50 


9-375319 


9 


11 


10-624681 


M 


22 


363954 


8 


85 


988073 


•5o 


37588i 


9 


6241 19 

623558 


23 


• 364485 


8 


84 


988043 


•5o 


376442 


9 


34 


37 


24 


365oi 6 


8 


83 


9 88oi3 


•5o 


377003 


9 


33 


622997 


36 


25 


365546 


8 


82 


987983 


•5o 


377563 


9 


32 


622437 


35 


26 


366075 


8 


81 


987953 


-50 


378122 


9 


3i 


621S78 


34 


27 


3666o4 


8 


80 


987922 
987892 


•5o 


378681 


9 


3o 


62 1319 


33 


28 


367 1 3 1 


8 


79 


-50 


379239 


9 


8 


620761 


32 


I 9 


367659 


8 


77 


987862 


-50 


379797 


9 


620203 


3i 


3o 


368 1 85 


8 


76 


987832 


•51 


38o354 


9 


27 


6 1 9646 


3o 


3i 


9-368711 


8 


75 


9-987801 


•51 


9-380910 


9 


26 


10-619090 
6i8534 


\l 


32 


369236 


8 


74 


987771 


-51 


38i466 


9 


25 


33 


369761 


8 


73 


987740 


•51 


382020 


9 


24 


6179S0 


27 


34 


370285 


8 


72 


987710 


•51 


382575 


9 


23 


617425 


26 


35 


370808 


8 


71 


987679 


•51 


383 1 29 


9 


22 


616871 


25 


36 


37i33o 


8 


70 


987649 


•5i 


383682 


9 


21 


6i63i8 


24 


U 


37i852 


8 


69 


987618 


•5i 


384234 


9 


20 


615766 


23 


372373 


8 


67 


987588 


•5i 


384786 


9 


IO 


6i52i4 


22 


39 


372894 


8 


66 


987557 


•5i 


385337 


9 


10 


6i4663 


21 


4o 


373414 


8 


65 


987526 


•5i 


385888 


9 


17 


614112 


20 


4i 


9-3 7 3 9 33 


8 


64 


9.987496 


•5i 


9-386438 


9 


i5 


io-6i3562 


\l 


42 


374452 


8 


63 


987465 


•5i 


386987 


9 


14 


6i3oi3 


43 


374970 


8 


62 


987434 


•5i 


387336 


9 


i3 


612464 


•7 


44 


375487 


8 


61 


987403 


•52 


388084 


9 


12 


611916 


16 


45 


376003 


8 


60 


987372 


•52 


38863i 


9 


1 1 


6n36 9 


i5 


46 


376019 


8 


5? 


987341 


-52 


389118 


9 


10 


610822 


14 


47 


377035 


8 


987310 


•52 


389724 


9 


a 


610276 


i3 


48 


377549 


8 


57 


987279 


•52 


390270 


9 


609730 


12 


i 9 


378063 


8 


56 


987248 


•52 


3 9 o8i 5 


9 


07 


609185 
608640 


1 1 


5o 


378577 


8 


54 


987217 


•52 


391360 


9 


06 


IC 


5i 


9-379089 


8 


53 


9-987,86 


•52 


9-391903 


9 


o5 


10-608097 


\ 


52 


379601 
38ou3 


8 


52 


987155 


•52 


392447 


9 


04 


607553 


53 


8 


5i 


987124 


-52 


392989 


9 


o3 


60701 1 


7 


54 


380624 


8 


5o 


987092 


•52 


3 9 353i 


9 


02 


606469 


6 


55 


38u34 


8 


% 


987061 


•52 


394073 


9 


01 


605927 


5 


56 


38i643 


8 


9S7030 


.52 


394614 




00 


6o5386 


4 


£ 


382.52 


8 


47 


98699S 


•52 


3931 54 


8 


9Q 


604846 


3 


382661 


8 


46 


986967 


•52 


393694 


8 


98 


6o43 06 


2 


59 


383 1 68 


8 


45 


986936 


•52 


396233 


8 


97 


603767 


1 


6o 


383675 


8 


44 


986904 
Sine 


•52 

76^ 


396771 
Cotang. 


8 


96 


603209 





" 


Cosine 




D. 


1). 


Taiifi M 



25* 



32 


(14 DEGREES.) A 


TABLE OF LOGARITHMIC 




M. 




Sine 


D. 


Cosine | D. | Tang. 


D. 


Cotang. 


60 


9-3836 7 5 


8-44 


9-986904 -52 


9 -39677 1 


8-96 


IO-6o322C. 


i 


384182 


S-43 


986873. -53 


397301; 


8.96 


602691 
602164 


IS 


a 


384687 


8-42 


986841 -53 


397846 


8-95 


3 


385i 9 2 


8-41 


986809 1 -53 
986778! -53 


3 9 8383 


8-94 


601617 


57 


4 


3856 9 7 


8-4o 


398910 
399433 


8- 9 3 


601081 


56 


5 


386201 


8-3 9 


986746 -53 


8-92 


600545 


55 


6 


386704 


8-38 


9867 1 4 1 ■■& 


399990 


8-91 


600010 


54 


I 


387207 


S-3 7 


986683 -53, 400024 


8-90 
8-89 
8-88 


590476 


53 


387709 


8-36 


9 8665 1 


-53 


{oio58 


598942 


52 


rj 


3882 10 


8-35 


986619 


•53 


401591 


598409 


1 5i 


10 


3887 1 1 


8-34 


986587 


•53 


402124 


8-87 


597876 


5o 


II 


9389211 


8-33 


1 9-986555 


•53 


9-402656 


8-86 


10-597344 


% 


12 


38971 1 


8-3s 


9 86523 -53 


403187 
403718 


8-85 


5 9 68 1 3 


i3 


390210 


8-3i 


986491' '53 


8-84 


596282 


47 


14 


390708 


8-3o 


986439' -53 


404249 


8-83 


595751 


46 


i5 


391206 


8-28 


986427J -53 


404778 


8-82 


595222 


45 


16 


391703 


8.27 


986395. -53 4o53o8 


8-81 


594692 


44 


«7 


392199 


8-26 


986363 '54 


4o5836 


8-8o 


594164 


43 


1 8 


392695 


8- 2 5 


98633 1 1 -54 


4o6364 


8-70 
8-78 


593636 


42 


19 


393191 


8-24 


986299 -54 


406892 


D93 1 08 


41 


20 


393685 


8-23 


986266 -54 


407419 
9-407945 


8-77 
8.76 


592581 


40 


21 


9-394I79 


8-22 


9-986234 '54 


io-592o55 


3 9 


22 


394673 


8-21 


986202 -54 


408471 


8. 7 5 


591529 


38 


23 


395166 


8-20 


986169 -54 


408997 
409021 


8.74 


59 1 oo3 


37 


24 


3 9 5658 


8-19 
8.18 


986137 -54 


8-74 


590479 
589955 


36 


25 


39600 


986 1 041 -54 


4ioo45 


8- 7 3 


35 


26 


396641 


8.17 


986072 -54 


410569 


8-72 


58 9 43 1 


34 


a 


397132 


5"7 


986039 


•34 


41 1092 


8,7, 


588 9 o8 


33 


39-7621 


8.16 


986007 


•54 


4n6i5 


8-70 


588385 


32 


29 


3981 1 1 


8-i5 


985974 


•54 


412137 


8-69 
8-68 


58 7 863 


3i 


3o 


398600 


8.14 


985942 


•54 


412658 


58 7 342 


3o 


3i 


9-399088 


8-i3 


9-985909 
985876 


• 55 


9-413179 


8-67 


io-58682i 


3 


32 


399575 


8-12 


• 55 


413699 


8-66 


5863oi 


33 


400062 


8- n 


985843 


• 55 


414219 

414738 


8-65 


585 7 8i 


27 


34 


4oo54q 


8-io 


98581 1 


• 55 


8-64 


585262 


26 


35 


4oio35 


8-09 
8-o8 


9 85 77 8 


• 55 


413257 


8-64 


584743 


25 


36 


401 520 


985745 


• 55 


415775 


8-63 


584225 


24 


ll 


4o2oo5 


8-07 
8-o6 


985712 


• 55 


416293 


8-62 


583707 


23 


38 


402489 


9 856 79 -55 


416810 


8-6i 


583190 


22 


3 9 


402972 


8-o5 


9 85646l -55 


417326 


8-6o 


582674 


21 


4o 


4o3455 


8.04 


9856i3; «55 


417842 


8-5 9 
8 -,58 


582i58 


20 


4i 


9 403938 


8-o3 


9- 9 8558o, -55 


9 -4i 8358 


io-58i642 


\l 


42 


404420 


8-02 


985547 -55 


4i88 7 3 


8-57 


581127 


43 


404901 
4o5382 


8-oi 


9855i4| '55 


419387 


8-56 


58o6i 3 


17 


44 


8-oo 


980480! -55 


419901 


8-55 


580099 
57 9 585 


16 


45 


4o5862 


7-99 
7.98 


985447 -55 


42041 5 


8-55 


i5 


46 


4o634i 


985414 * 36 


420927 


8-54 


579073 


14 


% 


406820 


7-97 


9 8538o -56 


421440 


8-53 


578560 


i3 


407299 


7-96 


985347 -56 


421952 


8-52 


578048 


12 


49 


407777 


7- 9 5 


9 853i4! -56 


422463 


8-5i 


5 77 537 


II 


5o 


408254 


7-94 


985280 -56 


422974 


8-5o 


577026 


10 


.)i 


9 408731 


7-94 


9-985247! -56 


9-423484 


8-40 
8-48 


10-576516 


8 


5a 


409207 


7.93 


9 852i3 -56 


428093 


576007 


53 


409682 


7-92 


985 1 8o: -56 


4245o3 


8-48 


575497 
574989 


"» 


54 


410107 


7-91 


985146 -56 423011 


8-47 


6 


55 


4io632 


7-90 


9 85n3 -56 425519 


8-46 


574481 


5 


56 


411106 


7-88 


980079! -56 


426027 


8-45 


573973 


4 


11 


41 1579 


985043; -56 


426534 


8-44 


573466 


3 


4I2052| 


7.87 


985ou : -56 ! 427041 


8-43 


5 7 2 9 5o 
572453 


2 


5 9 


4i2524 ! 


7-86 


984978J ' 56 i 427347 


8-43 


1 


60 


41 2996 j 


7-85 


984944 


56 428052 


8.42 


571948; 




Cosine ' 


D 1 


Sine 


75°| 


Cotang. 


D. 


Tancr. 1 M.J 



m: 



SINES AND TANGENTS. (15 DEGREES.) 



S3 



9 

10 

II 
12 

i3 
14 
i5 
16 

«7 

18 

19 
20 
21 
22 

23 

24 

23 

26 

11 

II 

3i 

32 

33 
34 
35 
36 

u 

39 

40 
41 

42 
43 

44 
45 



8 

& 

5i 

52 

53 
54 
55 
56 



Suio 
9-412996 
413467 
4i3938 
414408 
414878 
4i5347 
4i58i5 
416283 
416751 
417217 
417684 
;-4i8i5o 
4i86i5 
419079 
419544 
420007 
420470 
42oo33 
421J95 
421857 
4223i8 
9-422778 
423238 
42^697 
424106 
424613 
425073 
42553o 
425987 
426443 
426899 
9-42733- 
427809 
428263 
428717 
429170 
429623 
430075 
43o527 
430978 
431429 
9-431879 
432329 
432778 
433226 
433675 
434122 
434569 
435oi6 
435462 
435qo8 
5*436353 
436798 
437242 
437686 
438129 
438572 
439014 
439456 
439897 
44o338 



Cosine 



_IX_ 

7-85 
7-84 
7-83 
7-83 



I Cosine 1 D. 



7 

7 

7 

7 

7 

7 

7 

7 

7 

7 

7 

7 

7 

7 
7' 

7' 
7' 
7- 
7- 
7-65 
7-64 
7-63 
7-62 
7.61 
7-60 
7-60 
7-5 9 
58 

57 
56 
55 
54 
53 

52 
52 

5i 

DO 



7-49 
7-49 
7-48 
7-47 



7-36 
7-36 
7-35 
7-34 



9-984944; 
984910' 

984876! 
984842 ! 
984808! 
984774 
984740 
084706 ! 
984672! 
984637I 
984603 



9845351 
984500] 
984466 
984432 
984397 
984363 
984328 
984294 
984209 
j- 984224 
984190 
984155 
984120 
984085 
984050 
98400 
983981 
983946 
98391 1 
(.983875 
983840 
9 838o5 
983770 
983730 
980700 
98366. 
983629 
983594 
9835o8 
•983523 
983487 
983452 
983416 
98338i 
983345 
983309 
983273 
983238 
983202 
•983166 
983 1 3o 
983094 
983oo8 
983022 
982986 
982950 
982914 
982878 
982842 



D. I 



bine 1 



V 

'V 
•37 

'57 
V 

57 
•57 

•57 
•57 
•57 

:U 

-58 
•58 
•58 
•58 

•58 
•58 

•58 
■53 
•58 
•58 
•58 
-58 
•58 
•58 
•58 

• 58 

I 9 
•o 9 

.59 

• 3 9, 

• 5 9 

• 5 9 
•5 9 
■59 
'?' 
.59 

'?9. 
09, 
.60 
6o| 
60 
60 
60 
60 
60 
6o| 
60 
60 
60 
60 
6o, 
74° 



Tang. 



I^TTT "^- .- 



i-428o52 
42855 7 
429062 
429066 
430070 
430073 
431075 
43 1 577 
432079 
43i58o 
433o8o 
•43358o 
434080 
434579 
435078 
435576 
436073 
436570 
437067 
43i563 
438o59 
438554 
439048 
439043 
44oo36 
440529 
441022 
44i5i4 
442006 
442497 
442988 
443479 
443968 
444458 
444947 
445435 
445923 
44641 1 
446898 
447384 
447870 
9-448356 
448841 
449326 
449810 
400294 
400777 
45i26o 
45i743 

452225 

452706 
9-453187 
453668 
404148 
454628 
455io7 
455586 
456o64 
456542 
457019 
457496 
Cotang. 



D. 



8-42 

8.41 

8.40 

8-3 9 

8-38 

8-38 

8-3 7 

8-36 

8-35 

8-34 

8-33 

8-32 

8-32 

8-3i 

8-3o 

8-29 

8-28 

8-28 

8-27 

8-26 

8-25 

8-24 

8-23 

8-23 

8-22 
8-21 
8-20 

8-19 
8-i 9 
8.18 
8.17 
8-i6 
8-16 
8-i5 
8-14 
8-i3 

8-12 
8-12 

8-n 
8-io 
8-09 
8-09 
8-o8 
8-07 
8-o6 
8-06 
8-o5 
8-04 
8-o3 

8-02 
8-02 

8-oi 
8-oo 
7-99 
7-99 
7.98 

7-97 
7-96 , 
7.96 1 

7-95 j 
7-94_ 
D. 



Cotang 



10-571948 
57U43 
570938 
57043 
569930 
569427 
568 9 25 
568423 
567921 
567420 
566920 
o- 566420 
565920 
565421 
564922 
564424 
563927 
56343o 
562933 
562437 
561941 
io-56i446 
560952 
560457 
559964 
559471 
558978 
558486 
557994 
5570o3 
057012 
10-556521 
556o32 
555542 
555o53 
554565 
554077 
553589 
553 1 02 
5526i6 
552i3o 
io-55i644 
55 1 1 59 
550674 
550190 
549706 
540223 
548740 
548207 
547775 
54729 
10 -5468 1 3 
546332 
545852 
540372 
544892 
544414 
543936 
543458 
542981 
542604 



1 T»h&_J!L 



60 

u 

n 

55 
54 
53 

52 

5i 
5o 

8 

47 

46 
45 
44 
43 
42 
41 
40 

12 

37 
36 
35 
34 
33 

32 

3i 
3o 

3 

27 
26 

25 
24 
23 
22 
21 
20 

13 

n 

16 
i5 
14 
i-3 
12 
11 
10 



34 


(16 


DEGREES.) A 


TABLE OF LOGARITHMIC 




M. 


Sine 


1). 


Cosine 


J). 


Tang. 


D. 


Cotang. 


"6o~ 





9-44o338 


7-34 


9-982842 


.60 


9-457496 


7-94 


io-54?5o4 


i 


440778 


7 


■ 33 


982805 


•6o 


457073 


7-93 


542027 


it 


2 


441218 


7 


32 


982769 


•61 


45844Q 
458926 


7.93 


54i55i 


3 


44i658 


7 


3i 


982733 


•61 


7-92 


541075 


57 


4 


442096 
442535 


7 


3i 


982696 


•61 


459400 


7-91 


540600 56 


5 


7 


3o 


982660 


■61 


459875 


7-9° 


540125 55 


6 


442973 


7 


29 


982624 


•61 


460349 


7-88 


539651! 54 


I 


4434IO 


7 


28 


982587 


•61 


460823 


539177 53 
538703! 5a 


443847 


7 


27 


982551 


•61 


461297 


9 


444284 


7 


27 


982514 


•61 


461770 


7.88 


53823o 5i 


10 


444720 


7 


26 


982477 


•61 


462242 


7-87 


537758, 5o 


n 


9-445i55 


7 


25 


0-982441 


•61 


9-462714 


7.86 


:a 537286 49 


12 


445590 


7 


24 


982404 


•61 


463 1 86 


7-85 


5368i4 


48 


i3 


446025 


7 


23 


982367 


•61 


463658 


7-85 


536342 


47 


14 


446459 
446893 


7 


23 


982331 


•61 


464129 


7-84 


5358 7 i 


46 


i5 


7 


22 


982294 
982267 


•61 


464599 


7-83 


5354oi 


45 


16 


447326 


7 


21 


■6l 


465069 


7-83 


53493i 


44 


17 


447759 




20 


982220 


•62 


46553o 


7.82 


53446i 


43 


18 


448191 


7 


20 


982183 


-62 


466008 


7.81 


533992 
533524 


42 


19 


448623 


7 


IO 


982146 


•62 


466476 


7.80 


41 


20 


449054 


7 


l8 


982109 


•62 


466945 


7.80 


533o55 


40 


21 


9-449485 


7 


17 


9-982072 


•62 


9-4674i3 


?3 


io-53258 7 


li 


22 


4499i5 


7 


16 


982035 


•62 


467880 
468347 


532120 


23 


45o345 


7 


16 


981998 


•62 


7-78 


53 1 653 


li 


24 


450775 


7 


i5 


981961 


-62 


468814 


I'll 


53n86 


. 25 


45 1 204 


7 


14 


981924 
981886 


•62 


469280 


7-76 


530720 


35 


26 


45i632 


7 


1 3 


•62 


469746 


7-75 


53o254 


34 


\l 


452o6o 


7 


i3 


981849 


•62 


4702 1 1 


7 . 7 5 


529789 


33 


452488 


7 


12 


981812 


•62 


470676 


7-74 


529324 


32 


\9 


45291 5 


7 


11 


981774 


•62 


471141 


7-73 


5 2 885 g 


3i 


3o 


453342 


7 


10 


981737 


•62 


471605 


7- 7 3 


5283 9 5 


3o 


3i 


9-453768 


7 


10 


9-981699 


•63 


9-472068 


7.72 


10-5279.32 


2 2 


32 


454194 


7 


3 


981662 


•63 


472532 


7.71 


527468 


28 


33 


454619 


7 


981625 


•63 


472995 
473457 


7.71 


527005 


27 


34 


455o44 


7 


07 


981587 


•63 


7.70 


526543 


26 


35 


455469 


7 


07 


981549 


•63 


473919 
47438i 


7 .6 9 


526081 


25 


36 


4558 9 3 


7 


06 


981512 


•63 


?:S 


5256iq 


24 


ll 


4563 1 6 


7 


o5 


981474 


•63 


474842 


525i58 


23 


456739 


7 


04 


981436 


•63 


4753o3 


7 .6 7 


524697 
524237 


22 


3 9 


457162 


7 


04 


981399 


•63 


475763 


7 -b 7 


21 


40 


457584 


7 


o3 


9 8i36i 


•63 


476223 


7.66 


523777 


20 


41 


9- 458oo6 


7 


02 


9-98i323 


•63 


9-476683 


7-65 


10.523317 


•9 


42 


458427 


7 


01 


981285 


• 63 


477 l 42 


7-65 


522858 


18 


43 


458848 


7 


01 


981247 


•63 


477601 


7-64 


522399 


'7 


44 


459268 


7 


00 


98,1 209 


•63 


478059 


7 .63 


52 1 94 1 


16 


45 


459688 


6 


$ 


981171 


• 63 


478517 


7-63 


52U83 


i5 


46 


460108 


6 


98u33 


•64 


478975 


7.62 


521025 


14 


47 


460527 


6 


98 


981095 
981007 


• 64 


479432 


7.61' 


520568 


i3 


48 


460946 
46 1 364 


6 


97 


.64 


479889 
48o345 


7.61 


520III 


12 


49 


6 


96 


981019 


• 64 


7.60 


519655 


1 1 


5o 


461782 


6 


95 


980981 


.64 


480801 


7 .5 9 


519199 

10-518743 


10 


5i 


9-462199 


6 


9 5 


9-980942 


• 64 


9-481257 


?:S 


t 


52 


462616 


6 


94 


980904 
980866 


• 64 


481712 


518288 


53 


463 o3 2 


6 


9 3 


• 64 


482167 


7-57 


5 i 7 833 


1 


54 


463448 


6 


93 


980827 


.64 


482621 


m 


517379 
516925 


6 


55 


463864 


6 


92 


980789 


.64 


483075 


5 


56 


464279 


6 


9 1 


980750 


.64 


483529 


7.55 


5 1 647 1 


4 


n 


464694 


6 


90 


980712 


-64 


483982 


7.55 


5i6oi8 


3 


465 1 08 


6 


90 


980673 


• 64 


484435 


7-54 


5 1 5565 


2 


59 


465522 


6 


& 


9 8o635 


-64 


484887 


7-53 


5i5n3 


1 


60 


465935 


6 


980596 


• 64 


48533 9 


7-53 


514661 





Cosine 


P. 


Sine 


73° 


Cotang. 


D. 


Tang. 





SrNEb AND TANGENTS. 


(17 DEGREES. 


) 


35 


XL 


Sino 


D. 


Cosine 


D. | Tan£. 


D. 


Cotang. 


_ zn 





9-465q35 


6-88 


9.980596 
98o558 


• 64 


9-485339 


7-55 


lo-5i466i 


5o i 


i 


466348 


6-88 


.64 


485791 


7-52 


514209 

5i3 7 58 


u 




2 


466761 


6-87 


980519 


65 


486242 


7 .5i 




3 


467173 


6-86 


980480 


• 65 


486693 


7 .5i 


5i33o7 


57 


4 


467585 


6-85 


980442 


•65 


487143 


7«5o 


512857 


56 




5 


467996 


6-85 


980403 


• 65 


487593 


7-49 


512407 


55 




6 


468407 


6-84 


980364 


•65 


488043 


7-4Q 
7-48 


5i 1957 


54 




7 


468817 


6-83 


98o325 


• 65 


488492 


5n5o8 


53 




8 


469227 


6-83 


980286 


•65 


488941 


7-47 


5uo59 


52 




9 


469637 


6-82 


980247 


•65 


489390 


7-47 


5io6io 


5i 




10 


470046 


6-8i 


980208 


• 65 


489838 


7.46 


5ioi62 


5o 




ii 


9'470455 


6-8o 


9-980169 


• 65 


9-490286 


7.46 


10-509714 


% 




12 


470863 


6-8o 


980130 


• 65 


490733 


7.45 


509267 
5o882o 
5o8373 




i3 


471271 
471679 


6-79 
6-78 


980091 
980002 


•65 

• 65 


49 1 1 80 
491627 


7-44 
7-44 


% 




i5 


472086 


6-78 


980012 


• 65 


492073 


7-43 


507927 


45 




16 


472492 


6-77 


979973 


.65 


492519 


7.43 


507481 


44 




12 


472898 


6.76 


979934 
979895 


• 66 


492965 


7-42 


5o7o35 


43 




4733o4 


6-76 


• 66 


493410 


7.41 


506590 


42 




19 


473710 


6-75 


979855 


•66 


493854 


7.40 


5o6i46 


4i 




?.o 


474i 1 5 


6-74 


979816 


• 66 


494299 
9-494743 


7.40 


5o57oi 


40 




21 


9'4745iq 

474923 


6-74 


9-979776 


.66 


7.40 


io-5o52D7 


39 




22 


6- 7 3 


979737 


• 66 


495i86 


]'M 


5o48i4 


38 




23 


475327 


6-72 


979697 


• 66 


49563o 


504370 


37 




24 


473730 


6-72 


979658 


• 66 


496073 


7-37 


503927 


36 




20 


476i33 


6.71 


979618 


• 66 


49651 5 


7-37 


5o3485 


35 




26 


476536 


6-70 


979579 


• 66 


496957 


7-36 


5o3o43 


34 




27 


476938 
477340 


6-69 


979539 


• 66 


497399 


7-36 


5o26oi 


33 




28 


6-69 
6-68 


979499 
979409 


• 66 


497841 


7-35 


5o2i5o 
501718 


32 




?9 


477741 


.66 


498282 


7-34 


3i 




3o 


478142 


6-67 


979420 


• 66 


498722 


7-34 


501278 


3o 




3i 


9-478542 


6-67 


9-979380 


• 66 


9-499 163 


7-33 


io-5oo837 


3 




32 


478942 


6-66 


979340 


• 66 


499603 


7-33 


5oo3o7 




33 


479342 


6-65 


979300 


.67 


5ooo42 


7-32 


499968 


27 




34 


479741 


6-65 


979260 


.67 


5oo48i 


7-3i 


499319 


26 




35 


480140 


6-64 


979220 


.67 


5oog2o 


7-3i 


499080 
498641 


25 




36 


480539 


6-63 


979180 


- 67 1 5oi359 


7-3o 


24 




3 7 


480937 
48i334 


6-63 


979 '4o 


.67, 5oi 79 7 

•67 502235 


7-3o 


498203 


23 




38 


6-62 


979100 


7-29 


497765 


22 




3 9 


481731 


6-6i 


979059 


•67J 502672 


7.28 


497328 


21 




4o 


482128 


6-6i 


979010 


•671 5o3i09 


7.28 


496891 


20 




41 


9-482525 


6-6o 


9-978979 


•671 9 5o3546 


7-27 


10-496404 


IO 




42 


482921 


6-5 9 


978939 


•67 503982 


7-27 


496018 


10 




43 


4833 1 6 


6-5 9 
6-58 


978898 


.67 


5o44i8 


7-26 


495582 


17 




44 


483712 


978838 


• 67 


5o4854 


7. 2 5 


495146 


16 




45 


484107 


6-57 


978817 


.67 


5o5289 


7-2D 


4947 1 1 


i5 




46 


4845oi 


6-57 


978777 


.67 


5o5724 


7-24 


494276 


14 




47 


484895 
485289 


6-56 


978736 


•67 


5o6i59 


7-24 


493841 


i3 




48 


6-55 


978696 
97 8655 


• 68 


506393 


7-23 


493407 


12 




t 9 


485682 


6-55 


• 68 


507027 


7-22 


492973 


1 1 




5o 
5i 


486075 


6-54 


978615 


• 68 


507460 


7-92 


492340 


10 




9-486467 


6-53 


9-978574 


-68 ! 9.507893 
-68 5o8326 


7-21 


ir 492107 


I 




52 


486860 


6-53 


978533 


7-21 


491674 




53 


48 7 25i 


6-5a 


9784Q3 


•68, 508759 


7-20 


491 241 


7 




54 


487643 


6-5i 


978452 


•68 509191 


7- 19 


490809 


6 




55 


488o34 


6-5i 


97841 1 


•68 509622 


?:a 


490378 


5 




56 


488424 


6-5o 


978370 


•68 5ioo54 


489946 
489515 


4 




^ 


488814 


6-5o 


978329 


•68 5io485 


7.18 


3 




58 


489204 


6-4Q 


978288 


•68 5io 9 i6 


7.16 


489c 84 


2 




^ 9 


489J93 
480982 


6- 48 


978247 


•68 5n346 


488654 


1 




60 


6-48 


978206 


•68 
72° 


511776 


7.16 


488224 









Conine 


D. 


Sine 


Cotang. 


_D. 


_Tan£ L _ 


.il a 



36 


(18 DEGREES.) A 


TABLE OF LOGARITHMIC 




M. 


Sine 


D. 


Cosine | 1). 


Tang. 


D. 


Cotang. 


60 





9-489982 
490371 


6-48 


9-978206' »68 


9-5i 1776 


7.16 


10 -488222! 


i 


6-48 


97 8i65! -68 


512206 


7.16 


487794 


58 


2 


490759 


6-47 


978124! -68 


5i2635 


7 -i5 


487365 


3 


491 147 


6-46 


978083, -69 


5 13064 


7-14 


486o36 
486O07 


57 


4 


49i535 


6-46 


978042! -69 


5 1 3493 


7'i4 


56 


5 


491922 
49230S 


6-45 


978001 1 -69 


D13921 

5i4349 


7-13 


486079 


55 


6 


6-44 


977959 -69 
977018, -69 


7 -i3 


48565i 


54 


7 


492695 
493081 


6-44 


5i4777 


7-12 


485223 


53 


8 


6-43 


977»77 


-69 


J15204 


7-12 


484796 


52 


9 


493466 


6-42 


977835 


.69 


5i563i 


7-u 


484369 
483 9 43 


5i 


IC 


49385 1 


6-42 


977794 
9. 977752 


•69 


5 1 6057 


7-10 


5d 


ii 


9-494236 


6.41 


-69 


9-516484 


7-10 


io-4835i6 


8 


12 


494621 


6-41 


9777 1 1 


.69 


516910 
5i 7 335 


7-09 


483090 


i3 


495oo5 


6-40 


977660 
977628 


-69 


7-09 
7-08 


482665 


47 


14 


495388 


6-3 9 


•69 


517761 
5i8i85 


482239 
48181O 


46 


i5 


495772 


6-3 9 
6-38 


977086 


•69 


7-08 


45 


16 


496154 


977544 


•70 


5i86i 


7-07 


481390 


44 


'7 


496537 


6-37 


977503 


•70 


519034 


7-06 


480966 


43 


18 


496919 
497001 


6-3 7 


977461 


•70 


519458 


7-06 


480042 


42 


19 


6-36 


977419 


•70 


5i 9 88 2 


7-o5 


480118 


4i 


20 


497682 


6-36 


977377 
9-977335 


•70 


52o3o5 


7-o5 


479695 


40 


21 


9-498064 


6-35 


•70 


9-520728 


7-04 


10-479272 


3 9 


22 


498444 


6-34 


977293 


•70 


52ii5t 


7-o3 


470849 


38 


23 


498825 


6-34 


977201 


•70 


52i573 


7-o3 


478427 


37 


14 


499204 


6-33 


977209 


•70 


521995 


7-o3 


478005 


36 


»5 


499084 


6-32 


977167 


•70 


522417 


7-02 


477583 


35 


(6 


499963 


6-32 


977125 


•70 


522838 


7-02 


477162 


34 


n 


5oo342 


6-3i 


977083 


•70 


52325q 


7.01 


476741 


33 


18 


500721 


6-3i 


977041 


•7° 


52368o, 


7-01 


476320 


32 


>9 


501099 


6-3o 


976999 


•70 


524100 


7-00 


475900 


3i 


lo 


501476 


6-29 


976907 


•70 


524020 


6-99 


475480 


3o 


ti 


9-5oi854 


6-29 
6-28 


9-976914 
976872 
97683o 


•70 


9-524939 


6.90 
6.98 


10-475061 


3 


\2 


D0223l 


•71 


52535 


474641 


53 


502607 


6-28 


•71 


520778 


6-98 


474222 


27 


U 


502984 


6-27 


976787 


*7' 


52619-7; 


6.97 


4738o3 


26 


)5 


5o336o 


6-26 


976745 


•71 


5266i5 


6-97 


473385 


25 


16 


5o3735 


6-26 


976702 


•71 


527033 


6.96 


472967 
472049 


24 


*7 


5o4i 10 


6-25 


976660 


'V 


52745i! 


6-96 


23 


58 


5o4485 


6-25 


976617 


•7' 


5278681 


6- 9 5 


472132 


22 


*9 


504860 


6-24 


976074 


•71 


528285 


6- 9 5 


47i7i5 


21 


4o 


5o5234 


6-23 


976532 


•71 


528702 


6-94 


471298 


20 


4i 


9-5o56o8 


6-23 


9-976489 


•7' 


9-529119' 


6- 9 3 


10-470881 


'2 


i2 


5o5 9 8i 
5o6354 


6-22 


976446 


•7' 


529035 


6- 9 3 


470465 


18 


43 


6-22 


976404 


•7 l 


029950 


6- 9 3 


470050 


'7 


44 


506727 


6-21 


976361 


•7i 


53o366 


6-92 


469634 


16 


45 


507099 


6-20 


6 7 63i8 -71 


530781 


6-91 


469219 


i5 


46 


507471 


6-20 


g-]62-j5 j -71 


531196 


6.91 


468804 


14 


47 


5o 7 843 


6.19 


9762321 -72 


53i6n 


6-90 


46838o 
467970 


i3 


48 


5o82i4 


6-19 


976189; -72 


532025 


6.90 


12 


49 


5o8585 


6-i8 


976146- -72 


532439' 
532853 


6- £9 


467561 


11 


5o 


5o8o56 
9-509326 


6.18 ! 


976io3 i -72 


6-89 


467147 


10 


5i 


6-i 7 : 


9- 976060' -72 


9 533266 


6-88 


10-466734 


I 


5s 


509696 


6-i6 j 


976017! -72 


533679 


6-88 


466321 


53 


5ioo65 


6-i6 I 


975974 .72 


534092 


6-87 


465908 


7 


54 


5io434 


6-i5 


975930; .72 
97 588 7 .72 


5345o4 


6-87 


4654o6 


6 


55 


5io8o3 


6-i5 


5349161 
535328 


6-86 


465o84 


5 


56 


5i 1 172 


6-14 


975844J -72 


6-86 


464672 


4 


2 


5u54o 


6-i3 


975800! -72 


535 7 3 9 ! 


6-85 


464261 


3 


511907 


6-i3 


975757 1 -72 


536i5o< 


6-85 


46385o 


2 


5 9 


612275 


6-12 


975714 .72 


53656i i 


6-84 


46343a 
463028 


1 


6o 


512642 


6-12 


975670 -72 


536972! 


6-84 







Cosino 


_©.__ 


Sine J71^°_ 


Co tang. 1 


D. _l 


Tang.__ 


mT 





BINES AND TANGENTS. 


(19 DEGREES. 


) 


3 

1 


M. 


Sine 


D. 


Cosine | D. 


Tang. 


1 D - 


Cotang. 


o 


9; 5 1 2642 


6-12 


9-975670 .73 


9-536972 


6-84 


io-463o28 60 


i 


5i3ooq 


6- 11 


975627 .73 


53 7 38 2 


6-83 


462618 


fg 


2 


5i337$ 


6 11 


975583 .73 


537792 
538202 


6-83 


462208 


3 


5i374i 


6-io 


975539 


•73 


6-82 


461798 


57 


4 


514107 


6-09 


975490 


•73 


5386i 1 


6-82 


461389 


56 


5 


5i4472 


6.09 
6-oB 


975452 .73 


539020 


6-8i 


460980 


55 


6 


5i4837 


975408! .73 
97 5365 .73 


539429 


6- 81 


460371 


54 


I 


5l5202 


6-o8 


53 9 837 


6-8o 


460163 


53 


5i5566 


6-07 


97 532i .73 
9752771 -73 


540245 


6-8o 


459755 


52 


9 


5 1 5930 


6-07 
6-o6 


54o653 


6-79 


45o347 
438939 


5i 


10 


516294 


975233 .73 


541061 


6- 7 Q 
6-78 


5o 


ii 


9.516637 


6-o5 


9-975189 .73 
975145, -73 


9.541468 


10-458532 


% 


12 


517020 


6-o5 


541875 


6-78 


458i25 


i3 


517382 


5-o4 


975101 .73 


542281 


6-77 


457719 


47 


14 


517745 


6-o4 


9 7 5o57 .73 


542688 


6-77 


4573i2 


46 


i5 


518107 


6-o3 


975oi3 


•73 


543094 


6.76 


456906 


45 


16 


518468 


6-o3 


974969 
974925 


•74 


543499 


6-76 


4563oi 


44 


n 


518829 


6-02 


•74 


543900 
5443 1 


6-75 


436095 


43 


18 


519190 


6-oi 


974880 


•74 


6- 7 5 


455690 
455285 


42 


'9 


5i955i 


6-oi 


974836 


•74 


544715 


6-74 


4i 


20 


51991 1 


6-oo 


974792 


•74 


545119 


6-74 


45488i 


40 


21 


9.520271 


6-oo 


9-974748 


•74 


9-545524 


6.73 


10-454476 


ll 


22 


52o63i 


5-99 


974703 


•74 


545928 


6-73 


454072 


23 


520990 


5.99 


974659 


•74 


54633i 


6.72 


453669 
453263 


37 


24 


52i349 


5- 9 8 


974614 


•74 


546735 


6-72 


36 


25 


521707 


5- 9 8 


974570 


•74 


547 1 38 


6-71 


452862 


35 


26 


522066 


5-97 


97452D 


•74 


547540 


6-71 


45246o 


34 


27 


522424 


5.96 


974481 


•74 


547943 
548345 


6-70 


452057 


33 


28 


522781 


5- 9 6 


974436 


•74 


6.70 


45 1 655 


32 


?9 


5 2 3i38 


5- 9 5 


974391 


•74 


548747 


6-69 


45i253 


3i 


3o 


523495 


5- 9 5 


974347 


•75 


549149 


6-69 
6-68 


43o85i 


3o 


3i 


9-523852 


5-94 


9-974302 


•75 


9- J49550 


10-430430 


2 


32 


524208 


5- 9 4 


974257 


•75 


549921 


6-68 


45oo4o 
449648 


33 


524564 


5- 9 3 


974212 


•75 


55o352 


6-67 


27 


34 


524920 


5- 9 3 


974167 


•75 


55o752 


6-67 


449248 


26 


35 


526275 


5-92 


974122 


•75 


55i i52 


6-66 


448848 25 


36 


52563o 


5-91 


974077 


• 75 


55i552 


6-66 


448448 


24 


ll 


525o84 
526339 
52669J 


5-91 


974o32 


.75 


55l 9 32 


6-65 


448048 


23 


5-90 


973987 


-75 


55235i 


6-65 


447649 


22 


39 


5-90 
5-8 9 


973942 
973897 


• 7 5 


552750 


6-65 


44725o 


21 


4o 


527046 


- 7 5 


553i49 


6-64 


44685i 


20 


4i 


9-527400 


5-8 9 
5-88 


9-973832 


•75 


9-553548 


6-64 


IO-446452 


:? 


42 


527753 


973807 


• 7 5 


553946 


6-63 


446o54 


43 


528io5 


5-88 


973761 -75 


554344 


6-63 


445656 


17 


44 


528458 


5-87 


973716 .76 


554741 


6-62 


445259 


16 


45 


528810 


5-8 7 


973671) .76 


555i3 9 


6-62 


444861 


i5 


46 


529161 


5-86 


973625 .76 


555536 


6-6i 


444464 


14 


% 


5295i3 


5-86 


9735So| -76 


555933 
556J29 


6-61 


444067 


i3 


529864 


5-85 


973535 


•76 


6-6o 


443671 


12 


i 9 


53o2i5 


5-85 


973489 


•76 


556723 


6 60 


443275 


1 


5o 


53o565 


5.84 


973444 


•76 


5571 2 1 


6-59 


442870 . > 
10-442483 9 
442087 8 
441692 j 7 
441 298 S 6 


f 1 


9-53o9i5 


5.84 


9-973398 


.76 


9-557517 


6 5 9 


5a 


53i265 


5-83 


973352 


•76 


55791I 

5583o8' 


6-5 9 


53 


53i6i4 


5.82 


973307 


.76 


6-58 


54 


53 1 963 
53 2 3i2 


5.82 


973261 


.76 


538702 


6-58 


55 


5.8i 


9732i5 


•76 


559097. 


6-57 


44oqo3| 5 


56 


53266i 


5.8i 


973169 .76 


539491 
55 9 885: 


6-57 


440 5oo 
4401 1 5 


4 


tl 


533009 


5.8o 


978124 -76 


6-56 


3 


533357 


5.8o 


973078 .76 


560279! 
560673! 


6 56 


439721 
439327 
43S934 


2 


J 9 


533704 


i$ 


973o32| -77 


6-55 


1 


6o 


534o52 


972986! .77 


56 1 066 


6-55 





Cosine 


D. 


_Siiie [ 


7<P 


Cotang. j 


1) 


_TangVJ 


¥7 



38 



(20 DEGREES.) A TABLE OJT LOGARITHMIC 



|M. 


Sine 


D. 


Cosine 


|JX 


Tang. 


D. 


| Cotang. 





o 


9-534002 


5.78 


9-972986 


i m T 


9-56io6t 


6-55 


10-438934 60 


I 


534399 


5.77 


97294c 


•71 


56 1 45c 


6-54 


438o4 


, 5? 


2 


03474a 


5.77 


972894 


y -r 


56i85i 


6-54 


438 1 4c 


3 


535og2 
535438 


5. 77 


97284^ 


■11 


562244 


6-53 


437756 


57 


4 


5.76 


972802 


'11 


562636 


6-53 


437364 


i 56 


5 


530783 


5.76 


972755 


'11 


563028 


6-53 


436972 
436581 


55 


6 


536129 


5-75 


972709 
972663 


•77 


563419 


6-52 


54 


I 


536474 


5- 7 4 


•11 


5638i 1 


6-52 


436i8c 
435 79 £ 


53 


5368i8 


5-74 


972617 


•77 


564202 


6-5i 


52 


9 


537i63 


5-73 


97257c 


•77 


564592 

564983 

9-56537^ 


6-5i 


4354oS 


5i 


10 


537507 


5-73 


972524 


•77 


6-5o 


435oi7 


5c 


u 


9-53785i 


5-72 


9-972478 


3 


b-5o 


I0-43462- 


% 


12 


538 1 94 
538538 


5.72 


97243i 


565763 


6-49 


434237 


i3 


5.71 


972385 


■?2 


566 1 53 


6-49 


43 J 847 


47 


U 


53888o 


5.71 


972338 


.78 


566542 


6-49 
6-48 


433458J 46 


i5 


539223 


5-70 


972291 


.78 


566o32 
567320 


433o68 


45 


16 


53 9 565 


5-70 


972245 


•78 


6.48 


432680 


44 


!2 


539907 


5-6 9 


972198 


•78 


567709 


6-47 


432291 


43 


540249 


5-6 9 
5-68 


972161 


•78 


568o 9 8 
568486 


6-47 


43ioo2 
43r5i4 


42 


»9 


540590 


972105 


•78 


6-46 


4i 


20 


5409J 1 


5-68 


972o58 


' l8 o 


568873 


6-46 


431127 


4o 


21 


0-541272 


5-67 


9-972011 


•78 


9-069261 


6-45 


10-430739 


\l 


22 


54i6i3 


5-67 


97 1 964 


•78 


569648 


6-45 


43o352 


23 


541953 


5-66 


971917 


• l8 o 


570035 


6-45 


429965 
429O78 


37 


24 


542293 
542632 


5-66 


971870 


•78 


570422 


6-44 


36 


23 


5-65 


971823 


•78 


570809 
57119O 
57i58i 


6-44 


429191 


35 


26 


542971 


5-65 


971776 


.78 


6-43 


4288o5 


34 


11 


5433io 


5-64 


971729 


•79 


6-43 


428419 
428o33 


33 


543649 
543987 


5-64 


971682 


•79 


571067 
572352 


6-42 


32 


29 


5-63 


97i635 


•79 


6-42 


427648 


3i 


3o 


544325 


5-63 


971588 


•79 


572738 


6-42 


427262 


3o 


3i 


q- 544663 


5-62 


9-971540 


•79 


9-573i23 


6-4i 


10-426877 


3 


32 


545ooo 


5-62 


971493 


•79 


573507 


6-41 


426493 


33 


545338 


5- 61 


971446 


•79 


573892 


6-40 


426108 


27 


34 


545674 


5- 61 


97i3o8 
97J35i 


•79 


574276 


6-40 


425724 


26 


35 


546011 


5-6o 


•79 


574660 


6-39 


425340 


25 


36 


546347 


5-6o 


97i3o3 


•79 


575044 


6-3 9 


424o56 
424673 


24 


ll 


546683 


5-5 9 


971256 


•79 


575427 


6-3 9 
6-38 


23 


547019 


5-5 9 
5-58 


971208 


•79 


576810 


424190 


22 


3 9 


547354 


971161 


•79 


576193 


6-38 


423807 


21 


4o 


547689 


5-58 


97Hi3 


•79 


576576 


6-37 


423424 


20 


4i 


9-548024 


5-5 7 


9-971066 


-8o 


9-576958 
577341 


6-37 


io-423o4i 


\l 


42 


54835 9 ! 
5486 9 3; 


5-5 7 


971018 


-8o 


6-36 


422659 


43 


5-56 


970970 


-8o 


577723 
578104 


6-36 


422277 


n 


44 


549027 


5-56 


970022 


• 80 


6-36 


421896 


16 


45 


54936o 


5-55 


970874 


• 8o 


578486 


6-35 


421014 


i5 


46 


549693 


5-55 


970827 


-8o 


578867 


6-35 


42ii33 


14 


4 A 


55oo26 


5-54 


970779, 


• 8o 


579248 6-34 


420762 


i3 


48 


55o309 


5-54 


970731 -8o 


579629J 6-34 


420371 


12 


i 9 


500692 


5-53 


970683: -8o 


580009 6*34 


419991 


u 


5o 


55io24: 


5-53 


970635! -8o 


58o38 9 6-33 


41961 1 


10 


5i 


o-55i356 


5-52 


9-970586 -8o 


9-580769 6-33 


10-419231 

4i885i 


1 


52 


55i68 7 | 


5-52 


970538 -8o 


581149 
58i52§ 


6-32 


53 


552oi8' 


5-52 


970490 


-8o 


6-32 


418472 


I 


54 


552349 


5-5i 


970442 


.80 


581907 


6-32 


418093 


55 


55268o 


5-5i 


970394 


-8o 


582286 


6-3i 


4177U 


5 


56 


553oio 


5-5o 


970345 


• 8i 


582665 


6-3i 


417335 


4 


u 


553341, 


5-oo 


970297 


• 81 


583043 


6-3o 


416957 
416578 


3 


553670 


5-49 


970249 


•81 


583422 


6-3o 


2 


59 


554ooo 


5- 4Q 


970200 


•81 


5838oo 


. 6-29 


416200 


1 


60 


554329, 


5-48 


970152 


• 8i 


584177 


6-29 


4I5823 1 i 




Cobiiie i 


JO. 


Sine 


3f>° 


Cotang. D. 


Tang. | M. 1 





SINES 


AND TANGENTS. 


(21 DEUREES.] 




39 


M. 
o 


S3 no 


D. 


Cosine 


D. 


Tang. 


D. 


Cotang. 




9-554329 
554658 


5 


48 


9-970152 


"87 


9-384177 


6-29 


io-4i5823 


6c 


i 


5 


48 


970103 


.81 


584555 


6 


3 


4i5445 


5 9 


2 


554987 
5553 1 5 


5 


47 


970055 


.81 


584932 
585309 


6 


4i5o68] 58 


3 


5 


47 


970006 


.81 


6 


28 


414691! 57 


4 


555643 


5 


46 


969957 


.81 


585686 


6 


27 


4U3i4 56 


5 


555971 


5 


46 


969909 


.81 


586062 


6 


27 


413938! 55 


6 


556299 


5 


45 


969860 


• 8i 


586439 


6 


27 


4i356i| 54 


7 


556626 


5 


45 


96981 1 


• 8i 


586813 


6 


26 


4i3i85 


53 


8 


556g53 


5 


44 


969762 


.81 


587190 


6 


26 


412810 


52 


9 


557280 


5 


44 


969714 


.81 


587066 


6 


25 


412434 


5i 


10 


557606 


5 


43 


969665 


.81 


587941 


6 


25 


412009 


5o 


n 


9-557932 


5 


43 


9*969616 


.82 


9 -5883 16 


6 


25 


10-411684 


49 


12 


558258 


5 


43 


969067 


.82 


588691 


6 


24 


4n3o9 


48 


i3 


558583 


5 


42 


969018 


.82 


589066 


6 


24 


410934 


47 


U 


558909 


5 


42 


969469 


.82 


589440 


6 


23 


410060 


46 


i5 


55g234 


5 


4i 


969420 


.82 


58 9 8i4 


6 


23 


410186 


45 


16 


55o558 


5 


4i 


969370 


.82 


5 9 oi88 


6 


23 


409812 


44 


)l 


55 9 883 


5 


4o 


969321 


.82 


590062 


6 


22 


409438 


43 


560207 


5 


40 


969272 


• 82 


590935 


6 


22 


409065 


42 


*9 


56o53i 


5 


i 9 


969223 


.82 


59i3o8 


6 


22 


408692 


4i 


20 


56o855 


5 


ll 


969173 


• 82 


591681 


6 


21 


408319 


40 


21 


9-56ii78 


5 


9-969124 


.82 


9-592054 


6 


21 


10-407946 


H 


22 


56i5oi 


5 


38 


969073 


.82 


592426 


6 


20 


407074 


23 


561824 


5 


37 


969025 
968976 


.82 


592798 


6 


20 


407202 


I 1 


24 


562146 


5 


3 7 


.82 


593170 


6 


19 


406829 


36 


25 


562468 


5 


36 


968926 
968877 


•83 


593542 


6 


l 9 


4o6458 


35 


26 


562790 


5 


36 


•83 


593914 


6 


18 


406086 


34 


2 7 


563 1 1 2 


5 


36 


968827 


•83 


594285 


6 


18 


4057 1 5 


33 


28 


563433 


5 


35 


968777 


•83 


5g4656 


6 


18 


4o5344 


32 


2 9 


563755 


5 


35 


968728 


•83 


595027 


6 


17 


404973 


3i 


3o 


564075 


5 


34 


968678 


•83 


5 9 53 9 8 


6 


n 


404602 


3o 


3i 


9-564396 


5 


34 


9-968628 


•83 


9-595768 


6 


n 


10-404232 


2Q 


32 


564716 


5 


33 


968578 


•83 


5 9 6i38 


6 


16 


4o3862 


28 


33 


565o36 


5 


33 


9 685 2 8 


•83 


5 9 65o8 


6 


16 


403492 


27 


34 


565356 


5 


32 


968479 


•83 


596878 


6 


16 


4o3i22 


26 


35 


565676 


5 


32 


968429 


•83 


597247 


6 


i5 


402753 


25 


36 


565995 


5 


3i 


968379 


•83 


597616 


6 


i5 


402384 


24 


37 


5663U 


5 


3i 


968329 


•83 


597985 
5 9 8354 


6 


i5 


40201 5 


23 


38 


566632 


5 


3i 


968278 


•83 


6 


U 


401646 


22 


3 9 


566951 


5 


3o 


968228 


.84 


598722 


6 


14 


401278 


21 


4o 


567269 


5 


3o 


968178 


•84 


599091 


6 


i3 


400909 


20 


4i 


9-56 7 58 7 


5 


29 


9-968128 


.84 


9 -5-99459 


6 


i3 


10 -400041 


IO 


42 


567904 

568222 


5 


2 § 


968078 


• 84 


599827 


6 


i3 


400173 


l8 


43 


5 


28 


968027 


•84 


600194 


6 


12 


399806 


'7 


44 


56853g 


5 


28 


967977 


•84 


6oo562 


6 


12 


399438 


16 


45 


568856 


5 


28 


967927 
967876 


• 84 


600929 


6 


11 


399071 
398704 


i5 


46 


569172 


5 


27 


• 84 


601296 


6 


11 


14 


47 


569488 


5 


ll 


967826 


• 84 


601662 


6 


11 


3 9 8338 


i3 


48 


569804 


5 


967775 


• 84 


602029 


6 


10 


397971 


12 


49 


570120 


5 


26 


967725 


• 84 


602395 


6 


10 


397605 


11 


5o 


570435 


5 


25 


967674 


• 84 


602761 


6 


10 


397239 


10 


5i 


9-570751 


5 


25 


9-967624 


• 84 


9-6o3i27 


6 


09 


10-396873 


8 


%2 


571066 


5 


24 


967573 


• 84 


6o3493 
6o3858 


6 


09 


396507 


53 


57i38o 


5 


24 


967522 


.85 


6 


3 


396142 


1 


54 


571695 


5 


23 


967471 


.85 


604223 


6 


395777 


6 


55 


572069 


5 


23 


967421 


• 85 


6o4588 


6 


08 


395412 


5 


56 


5 7 2323 


5 


23 


967370 


•85 


604953 
6o53i7 


6 


07 


395047 


4 


& 


572636 


5 


22 


967319 


•85 


6 


07 


3 9 4683 


3 


5729D0 


5 


22 


967268 


•85 


6o5682 


6 


C7 


3 9 43i8 


2 


59 


5 7 3263 


5 


21 


967217 


•85 


606046 


6 


06 


393954 
393090 


1 


6o 


573575 


5-21 


967166 


•85 


606410 


6 


06 





Cosine 


D. 


Sine 


G80 


Cotang. 


D. 


~T^7~ 



26 



40 


(22 DEGREES.) A 


TABLE OP LOGARITHMIC 




M. 


Si no 


D. 


Cosine | D. 


Tan*. 


D. 


| Cotang. 







9-573575 
573888 


5-21 


9-967166 -85 


9 • 6064 1 


6- 06 


10-393590 


60 


i 


5-20 


9671 1 5 .85 


60677c 


6-o6 


393227 


5? 


2 


574200 


5-20 


967064 


.85 


607137 


6-o5 


3 9 2863 


3 


574512 


5-19 


9670 I C 


• 85 


607500 


6-o5 


392500 


57 


4 


574824 


5-19 


966961 


• 85 


607863 


6-04 


39213" 


56 


5 


570136 


5-19 
5-i8 


966910 
9 6685 g 
966808 


• 85 


608225 


6-04 


391775 


55 


6 


575447 


• 85 


6o8588 


6-04 


391412 


54 


7 


575708 


5.i8 


• 85 


608950 
609312 


6-o3 


391050 


53 


8 


576069 


5- 17 


966756 


• 86 


6-o3 


390688 


52 


9 


576379 
576689 


5.17 


966705 


• 86 


609674 


6-o3 


390326 
389964 


5i 


10 


5-i6 


9 66653 


• 86 


6ioo36 


6-02 


5o 


ii 


9-576999 
577309 
577618 


5-i6 


9-966602 


.86 


9-610397 
610739 


6-02 


10-389603 


% 


12 


5-i6 


96655o 


• 86 


6-02 


389241 
38888o 


i3 


5-i5 


966499 


• 86 


611120 


6-oi 


47 


14 


577927 


5-i5 


966447 


• 86 


61 1480 


6-oi 


388520 


46 


i5 


578236 


5-14 


966395 


.86 


611841 


6-oi 


388io 9 


45 


16 


578545 


5-i4 


966344 


.86 


612201 


6- 00 


387799 
387439 


44 


!Z 


5 7 8853 


5-i3 


966292 


.86 


6i256i 


6-00 


43 


579162 


5-i3 


966240 


.86 


612921 


6-oo 


387079 


42 


!9 


579470 


5-i3 


966188 


.86 


6i328i 


5-99 


386719 


41 


20 


579777 


5-12 


966136 


.86 


6i364i 


5-99 
5.98 


38635 9 


40 


21 


•y-58oo85 


5-12 


9-966085 


.87 


9-614000 


io-386ooo 


ll 


22 


580392 


5. II 


966o33 


.87 


6i435 9 


5- 9 8 


385641 


23 


580699 


5. II 


960981 


.87 


614718 


5- 9 8 


385282 


37 


24 


58 1 oo3 


5.11 


965928 
960876 


•87 


616077 


5-97 


384Q23 
384565 


36 


25 


58i3i2 


5- 10 


•87 


6i5435 


5-97 


35 


26 


58i6i8 


5-io 


965824 


.87 


6 i 5793 


5-97 


384207 


34 


27 


581924 


5.09 


965772 


.87 


6161O1 


5.96 


383849 


33 


28 


582229 


5.09 


965720 


•87 


616509 


5-96 


383491 


32 


o 9 


582535 


5-09 
5-o8 


9 65668 


.87 


616867 


5.96 


383i33 


3i 


3o 


582840 


9 656i5 


.87 


617224 


5- 9 5 


3S2776 


3o 


3i 


9-583145 


5.o8 


9 -965563 


.87 


9-617582 


5. 9 5 


10-382418 


3 


32 


583449 


5.07 


9655i 1 


.87 


617939 
618295 
6i8652 


5. 9 5 


382061 


33 


583754 


5.07 


965458 


.87 


5.94 


38i 7 o5 


2 


34 


584058 


5.o6 


965406 


:S 


5.94 


38i348 


35 


58436i 


5.o6 


9 65353 


619008 


5.94 


380992 
38o636 


25 


36 


584665 


5.o6 


9653oi 


.88 


619364 


5. 9 3 


24 


u 


584968 


5-o5 


965248 


.88 


619721 


5. 9 3 


380279 


23 


585272 


5-o5 


965195 


.88 


620076 


5. 9 3 


379924 
37 9 568 


22 


3 9 


585574 


5-o4 


965i43 


.88 


620432 


5-92 


21 


4o 


■ 5858 77 


5-04 


965090 
9-965007 


.88 


620787 


5-92 


379213 


20 


4i 


9-086179 


5-o3 


.88 


9-621142 


5-92 


10-378858 


\l 


42 


586482 


5-o3 


964984 


.88 


621497 
621852 


5.91 


3785o3 


43 


586783 


5-o3 


96493 1 
964879 


.88 


5.91 


378148 


17 


44 


587085 


5-02 


.88 


622207 


5.90 


377793 


16 


45 


58 7 386 


5-02 


964826 


.88 


62256i 


5-90 


377439 


i5 


46 


687688 


5oi 


964773 


.88 


622915 


5.90 


377085 


14 


% 


587989 


5-oi 


964719 


.88 


623269 
623623 


i'b 


376731 


i3 


588289 


5-oi 


964666 


.89 


5.89 


376377 


12 


P 


5885 9 o 


5-oo 


964613 


.89 


623076 


5.89 


376024 


11 


5o 


588890 


5«oo 


964560 


.89 


62433o 


5.88 


375670 


10 


5i 


9*589190 
589489 


4-99 


9-964507 


.89 


9-624683 


5.88 


10-375317 


8 


52 


4-99 


964454 


.89 625o36 
.89' 625388 


5.88 


374964 


53 


58 97 8o 
5 9 oo88 1 


4- 90 
4-98 


964400 


5.87 


374612 


1 


54 


964347 


.89 625 7 4i 


5.87 


374259 


6 


55 


590387 


4-98 


964294 


.89 626093 


5.87 


373907 
373555 


5 


56 


5 9 o686 


4-97 


964240 


.89! 626445 


5.86 


4 


u 


590984 


4-97 


964187 


.89 1 626797 


5.86 


3732o3 


3 


591 2 82 


4-97 


964i33 


•8 9 i 627149 


5-86 


372851 2 


& 


591580 


4-96 


964080 


.89 1 627501 


5-85 


372499 1 . 
372148! 


6o 


59187b' 


4-96 


964026 


•89 627802 


5-85 


1 


Cosine 


D. i 


Sine 


G7°, Coking, i 


D. 


Tang. J M^ 



SINES AND TANGENT8. (23 DEGREES.) 



41 



M. 


Sine 


1 D ' 


Cosine 


D. | Tung. 


1 D. 


| Cotanj;. 


1 


o 


9.591878 


. 4-96 


9-964026 


•89 9-627852 


5-85 


10-37214^ 


60 


i 


592176 


! 4-9 5 


963972 


.89! 628203 


5-85 


371797 


5 9 

58 


2 


592473 


4-93 


963919 
963865 


•89! 62855/j 


5-85 


371446 


3 


592770 


4-9 5 


•90J 628903 


5-84 


371095 


57 


4 


593067 
5 9 3363 


4.94 


963811 


| -90, 629255 


5-84 


370745 


56 


5 


4-94 


963757 


1 -go! 629606 


5-83 


370394 


55 


6 


593659 


4- 9 3 


963704 


•90! 629956 


5-83 


370044 


54 


I 


593955 


4- 9 3 


96363o 


•9c 63o3o6 


5-83 


36969/ 


53 


594251 


4-93 


963596 


.90 


63o656 


5-83 


36 9 344 


52 


9 


594547 
594842 


4-92 


963542 


• 90 


63ioo5 


5-82 


368995 


5i 


10 


4.92 


963488 


■9c 


63i355 


5-82 


368645 


5o 


ii 


9.590137 


4-9" 


9-963434 


■ 90 


9-631704 


5-82 


10-368296 


49 


12 


590432 


4-91 


963379 


.90 


632053 


5-8i 


367947 


48 


i3 


595727 


4-91 


963323 


■90 6324oi 


5- 81 


367399 


47 


14 


396021 


4.90 


963271 


•90J 63275o 


5-8i 


367230 


46 


i5 


5963i5 


4.90 
4.89 


963217 


.90 


633o 9 8 


5-8o 


366902 


45 


16 


596609 
596903 


963 1 63 


.90 


633447 


5-8o 


366353 


44 


\l 


4.89 


963 l 08 


•91 


633795 


5-8o 


3662o5 


43 


597196 


4.89 


963o54 


•91 


634U3 


5-79 


365857 


42 


*9 


597490 


4-88 


962999 


•91 


634490 


5.79 


3655 10 


41 


20 


597783 


4-88 


962943 


•91 


634838 


5.79 
5.78 


365i62 


40 


21 


9.598075 


4-8 7 


9-962890 
9 62836 


• 9 f 


9-635i85 


io-3648i5 


3 9 


22 


598368 


4-8 7 


• 9 ! 


635532 


5.78 


364468 


38 


23 


5 9 866o 


4-8 7 


962781 


.91 


635879 


5- 7 8 


364121 


ll 


24 


598932 


4-86 


962727 


•91 636226 


5.77 


363774 


25 


599244 


4-86 


962672 


•91 


636572 


5.77 


363428 


35 


26 


599336 


4-85 


962617 


•91 


636919 


5.77 


363o8i 


34 


27 


599827 


4-85 


962562 


•91 


637265 


5.77 


362735 


33 


28 


600118 


4-85 


962508 


•91 


63 7 6n 


5.76 


36238 9 


32 


29 


600409 


4.84 


962453 


•91 


637956 
638302 


5-76 


362044 


3i 


3o 


600700 


4-84 


962398 


•92 


5.76 


36i6 9 8 


3o 


3i 


9 . 600990 


4.84 


9-962343 


.92 


9-638647 


5.75 


io-36i353 


3 


32 


601280 


4-83 


962288 


•92 


638992 
6393J7 


5.75 


36 1 008 


33 


601370 


4-83 


962233 


•92 


5.73 


36o663 


27 


34 


601860 


4-82 


962178 


•92 


639682 


5.74 


36o3i8 


26 


35 


602 1 30 


4-82 


962123 


•92 


640027 


5-74 


359973 


25 


36 


602439 


4-82 


962067 


•92 640371 


5.74 


359629 


24 


ll 


602728 


4.81 


962012 


•92 640716 


5.73 


359284 


23 


6o3oi7 


4.81 


901957 


•92' 641060 


5- 7 3 


358940 


22 


3 9 


6o33o5 


4-8i 


961902 


•92J 641404 


5.73 


358596 


21 


40 


603394 


4-8o 


961846 


•92 641747 


5-72 


358253 


20 


41 


9 .6o3882 


4-8o 


9-961791 


•92, 9-642091 


5-72 


10-357909 


18 


4T 


604170 


4-79 


961735 


•92 642434 


5-72 


357566 


43 


604457 


4-79 


961680 


•9 2 , 642777 


5-72 


357223 


»7 


44 


6o4745 


4-79 


961624 


•g3 643120 


5.71 


35688o 


16 


45 


6o5o32 


4-78 


961569 


•93 643463 


5.71 


356537 


i5 


46 


6o53i9 


4-78 


9 6i5i3 


• 9 3 ; 643806 


5-71 


356194 


14 


s 


6o56o6 


4-78 


961458 


•93 644148 


5-70 


355852 


i3 


600892 


4-77 


961402 


•931 644490 


5-70 


3555io 


12 


49 


606179 


4-77 


961346 


•93 644832 


5-70 


355i68 


11 


5o 


606465 


4-76 


961290 


•93 645174 


5-69 


354826 


10 


5i 


9-606751 


4-76 


9-961235 


•93 9-6455i6 


5-69 


io.354484 


Q 


52 


607036 


4-76 


961179 
961123 


• 9 3; 645857 


5-69 


354U3 


8 


53 


607322 


4-75 


•931 646199 


5-69 
5-68 


3538oi 


I 


54 


607607 


4-75 


961067 


•q3' 646340 


35346o 


55 


607892 


4-74 


96101 1 


•93' 646881 


5-68 


353no 

352778 


5 


56 


608177 


4-74 


960955 


•93 647222 


5-68 


i 


U 


608461 


4-74 


960899 
960843 


•93 647562 


5-67 


352438 


i 


608745 


4- 7 3 


•94' 647903 


5-67 


352097 


2 


5 9 


60Q029 
609313 


4- 7 3 


960786 


•94! 648243 


5-6 7 


351737 


1 


60 


4-73 


960730 


•94 648583 


5-66 


35i4i7 





1 


Cosine 


I). 


Sine 


GG C I Cotang. 1 


D. 


""rltn"^" 



42 


(24 


DEQREB8.) A 


rABLK OF LOOARITHMTO 




M. 

o 


Sino 


D. 


Cosine 


D. 


Tang. 


D. 


Cotang. 


60 


9»6o93i3 


4-73 


9-960730 


•94 


9-648583 


5-66 


io-35i4i7 


2 


609597 
609880 


4 
4 


72 
72 


960674 
960618 


•94 
•94 


648923 
649263 


5 
5 


66 

66 


351077 
350737 


is 


3 


610164 


4 


72 


960561 


•94 


649602 


5 


66 


350398 


57 


4 


610447 


4 


7* 


96o5o5 


.94 


649942 


5 


65 


35oo58 


56 


5 


610729 


4 


7i 


960448 


.94 


65o28i 


5 


65 


3497>9 


55 


6 


611012 


4 


70 


960392 


•94 


65o62o 


5 


65 


34938o 


54 


I 


61 1294 


4 


70 


96o335 


.94 


650959 


5 


64 


349041 
348703 


53 


61 1576 


4 


70 


960279 


.94 


651297 
65i636 


5 


64 


52 


9 


6u858 


4 


69 


960222 


.94 


5 


64 


348364 


5i 


IC 


612140 


4 


69 


960165 


■94 


651974 


5 


63 


348026 


5o 


ii 


9.612421 


4 


69 


9-960109 


- 9 5 


9*6523i2 


5 


63 


10-347688 


% 


is 


612702 


4 


68 


960052 


.95 


65265o 


5 


63 


34735o 


i3 


612983 


4 


68 


939995 


.95 


652988 


5 


63 


347012 


47 


U 


6i3264 


4 


67 


959938 


.95 


653326 


5 


62 


346674 


46 


i'i 


6i3545 


4 


67 


959882 


.95 


653663 


5 


62 


346337 


45 


ri 


6i3825 


4 


67 


959825 


.95 


654000 


5 


62 


346000 


44 


»! 


6i4io5 


4 


66 


959768 


.95 


654337 


5 


6i 


345663 


43 


iB 


6i4385 


4 


66 


9597 1 1 


.95 


654674 


5 


61 


345326 


42 


»9 


614665 


4 


66 


959654 


.95 


655ou 


5 


61 


344989 


4i 


20 


614944 


4 


65 


959596 
9-959539 


.95 


655348 


5 


61 


344652 


4o 


21 


9-6i5223 


4 


65 


.95 


9-655684 


5 


60 


io-3443i6 


is 


2J 


6i?5o2 


4 


65 


959482 


.95 


656o2o 


5 


60 


343980 


23 


6i5 7 8i 


4 


64 


959425 


.95 


656356 


5 


60 


343644 


37 


24 


616060 


4 


64 


9D9368 


.95 


656692 


5 


5 9 


3433o8 


36 


23 


6i6338 


4 


64 


95q3io 


.96 


657028 


5 


5 9 


342972 


35 


26 


616616 


4 


63 


959253 


.96 


657364 


5 


5 9 


342636 


34 


27 


616894 


4 


63 


959195 


.96 


657699 
658o34 


5 


5 9 


3423oi 


33 


28 


617172 

617450 


4 


62 


9D9138 


.96 


5 


-58 


341966 


3"2 


29 


4 


62 


959081 


.96 


658369 


5 


58 


34i63i 


3i 


3o 


617727 


4 


62 


939023 


.96 


658704 


5 


58 


341296 


3o 


3i 


9-618004 


4 


61 


9-958965 


.96 


9-659o3o 
65937J 


5 


58 


10-340961 


it 


32 


618281 


4 


61 


958908 


.96 


5 


5 7 


340627 


33 


6i8558 


4 


61 


95885o 


.96 


659708 


5 


57 


340292 
33 9 958 


27 


34 


6i8834 


4 


60 


938792 


.96 


660042 


5 


5 7 


26 


35 


6iqiio 


4 


60 


938734 


.96 


660376 


5 


5 7 


339624 


25 


36 


619386 


4 


60 


938677 


.96 


6607 1 


5 


56 


339290 
338967 


24 


3i 


619662 


4 


5 9 


958619 


.96 


661043 


5 


56 


23 


38 


619938 


4 


J 9 


938361 


.96I 661377 


5 


56 


338623 


22 


3g 


6202l3 


4 


& 


9385o3 


.97 661710 


5 


55 


338290 


21 


. 4o 


620488 


4 


938443 


•07! 662043 
7 


5 


55 


337967 


20 


4i 


9-620763 


4 


58 


9-958387 


•97 


9-662376 


5 


55 


10-337624 


» 


42 


62io38 


4 


5 7 


95832 9 


•97 


662709 


5 


54 


337291 
336958 


43 


62i3i3 


4 


57 


958271 


•97 


663o42 


5 


54 


1*7 


44 


621587 


4 


5 7 


9582i3 


•97 


6633 7 5 


5 


54 


336625 


l6 


45 


621861 


4 


56 


958 1 54 


•97 


663707 


5 


54 


3362 9 3 


i5 


46 


622135 


4 


56 


958096 


•97 


664039 


5 


53 


335 9 6i 


14 


47 


622409 


4 


56 


958o38 


•97 


664371 


5 


53 


335629 


i3 


48 


622682 


4 


55 


957979 


•97 


664703 


5 


53 


335297 


12 


49 


622956 


4 


55 


957921 


•97 


665o35 


5 


53 


334965 


1 1 


5o 


623229 


4 


55 


95 7 863 


•97 


665366 


5 


52 


334634 


10 


5i 


9-6235o2 


4 


54 


9-957804 


•97 


9 -6656 9 7 


5 


52 


io-3343o3 


i 


52 


623774 


4 


54 


957746 


.98 
.98 


666029 


5 


52 


333971 


53 


* 624047 


4 


54 


957687 


66636o 


5 


5i 


333640 


7 


54 


624319 


4 


53 


957628 


.98 


666691 


5 


5i 


333309 


6 


55 


624591 


4 


53 


957570 


.98 


667021 


5 


5i 


332979 
332648 


5 


56 


624863 


4 


53 


93751 1 


• 9 8, 667352 


5 


5i 


4 


U 


625i35 


4 


52 


957452 


• 9 8 ; 667682 


5 


5o 


3323i8 


3 


625406 


4 


52 


957393 


.98 668oi3 


5 


5o 


331987 


2 


5 9 


625677 


4 


52 


957335 


- 9 8 ! 668343 


5 


5o 


33i65t 


1 


6o 
L 


625948 
Cosine 


4 


5i 


937276 
Sine 


. 9 3 ; 668672 


5 5o 


33 1 328 





] 


[>. 


«35=> 


Cotang. 


~] 


0. 


Taru£.__ 



SINES AND TANGENTS. (25 DEGREES.) 



43 



M. 




Sino 


1). 


Cosine | D. 


Tang. 


L>. 


Cotimg'. 
io-33i327 
33009S 




9-6a5g48 


4-5i 


9.957276 .98 


9.668673 


5-5o 


60 


i 


626219 


4 


5i 


957217; 


.98 


669002 


5 


49 


U 


2 


626/490 


4 


5i 


9 5 7 i58 


. 9 8 


66 9 332 


5 


49 


33o668 


3 


626760 


4 


5o 


957099 


• 9 8 


669661 


5 


% 


33d3.?9 


5 7 


4 


627030 


4 


5o 


95 7 040 ; 


.98 


669991 


5 


330009 


56 


5 


627300 


4 


5o 


956981 


.98 


670320 


5 


48 


329680 


55 


6 


627570 


4 


49 


966921 
9 56862 


•99 


670649 


5 


48 


329351 54 


I 


627840 
628109 


4 


49 


•99 


67097] 
671306 


5 


48 


329023 53 


4 


49 


9368o3 


•99 


5 


47 


328694! 52 


9 


628378 


4 


48 


956744 


•99 


67 1 634 


5 


47 


328366 


5i 


IO 


628647 


4 


48 


9 56684 


•99 


671963 


5 


47 


328037 


5o 


ii 


9-628916 


4 


47 


g. 936625 


•99 


9-672291 


5 


47 


,0.327709 


% 


12 


629185 


4 


47 


936566 


•99 


672619 


5 


46 


32738i 


i3 


629453 


4 


47 


9565o6 


•99 


672947 


5 


46 


32 7 o53 
326726 


47 


14 


629721 


4 


46 


936447 


•99 


673274 


S 


46 


46 


i5 


629989 


4 


46 


9D6387 


•99 


673602 


5 


46 


3263 9 8 


45 


16 


630257 


4 


46 


956327 


•99 


673929 


5 


45 


326071 


44 


H 


63o524 


4 


46 


9 56268| 


•99 


674257 


5 


45 


325743 


43 


18 


630792 


4 


45 


9362081 


•00 


6 7 4584 


5 


45 


3254i6 


42 


x 9 


63 1 039 


4 


45 


936148 i 


•00 


674910 


5 


44 


325090 


41 


20 


■ 63i326 


4 


45 


936089 1 


•00 


673237 


5 


44 


324763 


40 


21 


9-63i5g3 


4 


44 


9-936029 1 


• 00 


0.673504 


5 


44 


I0 . 324436 


3 9 


22 


63i85o 

632123 


4 


44 


955969 i 


■00 


675890 


5 


44 


324110 


38 


23 


4 


44 


955909 1 


•00 


676216 


5 


43 


323784 


37 


24 


632392 


4 


43 


955849 i 


•00 


676543 


5 


43 


323457 


36 


25 


632638 


4 


43 


933789! i 


•00 


676869 


5 


43 


323i3i 


35 


26 


632923 


4 


43 


933729'] 


•00 


677194 


5 


43 


322806 


34 


27 


633 1 89 


4 


42 


955669'! 


•00 


677320 


5 


42 


322480 


33 


28 


633434 


4 


42 


955609 1 


■00 


677846 


5 


42 


322i54 


32 


29 


633719 


4 


42 


95554811 


•00 


678171 


5 


42 


321829 


3i 


3o 


633984 


4 


4i 


955488! 1 


•00 


678496 


5 


42 


32i5o4 


3o 


3i 


9-634249 


4 


4i 


9.955428 1 


•01 


9.678821 


5 


4i 


10.321179 


29 


32 


6343i4 


4 


40 


955368! 1 


•01 


679146 


5 


4i 


320854 


28 


33 


634778 


4 


4o 


955307 1 


•01 


679471 


5 


4i 


320529 


27 


34 


635o42 


4 


40 


955247 1 


•01 


679795 


5 


4i 


320205 


26 


35 


6353o6 


4 


3 9 


955 186! 1 


•0, 


680120 


5 


40 


319880 


25 


36 


63)370 


4 


3 9 


9DD 1 26i J 


•0. 


68o444 


5 


40 


319556 


24 


3 2 


635834 


4 


3 2 


955o65 1 


•01 


680768 


5 


40 


319232 


23 


38 


636097 


4 


38 


955oo5 1 


•01 


681092 


5 


40 


318908 
3 1 8584 


22 


3 9 


636360 


4 


38 


954944 1 
954883 1 


•01 


681416 


5 


39 


21 


4o 


636623 


4 


38 


•01 


681740 


5 


39 


318260 


20 


4i 


9-636886 


4 


37 


9-954823 1 


•01 


9.682063 


5 


39 


io.3i79 3 7 


IQ 


42 


637148 


4 


37 


954762 1 


•01 


68 2 387 


5 


3 9 


317613 


l8 


43 


63741 1 


4 


37 


954701 1 


•0. 


6S2710 


5 


38 


317290 


17 


44 


637673 


4 


37 


954640 i 


•01 


683o33 


5 


38 


316967 


16 


45 


637 9 35 


4 


36 


954579 1 
9 545i8 1 


•c: 


683356 


5 


38 


316644 


i5 


46 


638i 9 7 
638438 


4 


36 


•02 


683679 


5 


38 


3i632i 


14 


47 


4 


36 


954457 1 


• 02 


6S400 1 


5 


37 


315999 


i3 


48 


638720 


4 


35 


954396 1 


•02 


6H4324 


5 


37 


315676 


12 


49 


638981 


4 


35 


954335 1 


•02 


684646 


5- 


37 


3 1 5354 


11 


5o 


639242 


4 


35 


954274 ' 


•02 


684968 


5- 


37 


3i5o32 


10 


5i 


9-6395o3 


4 


34 


9-934213 1 


■02 


9-685290 


5 


36 


jr>.3l47IO 


9 


52 


639764 


4 


34 


954i52 1 


■02 


6856 1 2 


5 


36 


3U388 


8 


53 


640024 


4 


34 


954090 1 


•Oi 


685 9 34 


5 


36 


i 1 4066 


7 


54 


640284 


4 


33 


954029 1 


•02 


686255 


5- 


36 


3 1 3745 


6 


55 


640544 


4 


33 


9 53 9 68 1 


•02 


6865 77 


5 


35 


3i3423 


5 


56 


640804 


4 


33 


953906 1 


•02 


686898 


5- 


35 


3i3io2 


4 


57 


641064 


4 


32 


953845 1 


•02 


687219 


5- 


35 


312781 


3 


58 


641324 


4 


32 


953783 1 


•02 


687540 


5- 


35 


3 1 2460 


2 


5 9 


641584 


4 


32 


953722 1 


•o3 


68-1861 


5- 


34 


3i 2 139' 1 

3u8i8| 


60 


641842 


4-3i 


95366o 1 


•o3 


688182 


5- 


3/ 




Cosine 


J 


=^_. 


Sine [C 


M°, 


J 


5 \ 


Tang. 1 M. 



14 


(26 


DEGREES.) A TABLE OF LOGARITHMIC 




M. 




Sine 


D. 


Cosine 


i~o3 


Tang. 


D. 


Cotnng. 




9-641842 


4-3i 


9-953660 


9.688182 


5.34 


io-3ii8i8 


60 


i 


642101 


4-3i 


933599 
953537 


i-o3 


688502 


5 


•34 


3 1 1498 


% 


2 


64236o 


4-3i 


i-o3 


688823 


5 


•34 


311177 


3 


642618 


4-3o 


933473 


i-o3 


689143 


5 


•33 


310837 


57 


4 


642877 


4-3o 


9534i3 


i-o3 


689463 


5 


• 33 


3io537 


56 


5 


643 1 35 


4-3-0 


953352 


i-o3 


689783 


5 


-33 


310217 


55 


6 


643393 


4-3o 


953290 
q53228 


i-o3 


690103 


5 


•33 


3098971 54 


I 


64365o 


4-29 


i-o3 


690423 


5 


■ 33 


309377, 53 
3oo258' 52 
3o8 9 38: 5i 


643908 


4-29 


q33i66 


i-o3 


690742 


5 


•32 


9 


644i65 


4-29 


q53io4 


i-o3 


691062 


5 


•32 


m 


644423 


4-28 


933042 


1 -o3 


691381 


5 


-32 


308619! 5o 


1 1 


9 • 644680 


4 28 


9-952980 


1.04 


9-691700 


5 


• 3i 


io-3o83oo 


% 


12 


644936 


4 28 


932918 


[•04 


692019 


5 


■ 3i 


307981 


.3 


640193 


4 27 


932855 


1-04 


6 9 2338 


5 


•3i 


307662 


47 


H 


645400 


4 27 


932793 


[•04 


692656 


5 


3i 


307344 


46 


i5 


643706 


4 27 


952731 


[■04 


692975 


5 


3i 


307025 


45 


16 


643962 


4 26 


932669 


1-04 


693293 


5 


3o 


306707 
3o6388 


44 


\l 


646218 


4 26 


9526061 


[•04 


693612 


5 


3o 


43 


646474 


4-26 


952544 


1.04 


693930 


5 


3o 


306070 


42 


»9 


646729 


4-25 


932481! 


[•04 


694248 


5 


3o 


3o5752 


41 


20 


646984 


4-25 


932419! 


[•04 


694566 


5 


29 


3o543 i 


40 


21 


9-647240 


4-25 


9-932356 


[•04 


9-694883 


5 


29 


io-3o5iJ7 


3 9 


22 


647494 


4-24 


952294 


[•04 


695201 


5 


29 


3o4799 


38 


23 


647749 


4-24 


95223i 


• 04 


6 9 55i8 


5 


29 


304482 


37 


24 


648004 


4-24 


952168 


-o5 


6 9 5836 


5 


2 


304164 


36 


25 


648238 


4-24 


932106' 


• o5 


696153 


5 


3o3847 


35 


26 


648312 


4-23 


952043 


•o5 


696470 


5 


28 


3o353o 


34 


27 


648766 


4-23 


9 5i 9 <so; 


i-o5 


696787 


5 


■ 28 


3o32i3 


33 


28 


649020 


4-23 


931917 


i-o5 


697103 


5 


28 


302897 


32 


29 


649274 


4-22 


95i854| 


i-o5 


697420 


5 


27 


3o258o 


3i 


3o 


649327 


4-22 


951791! 


• o5 


697736 


5 


27 


302264 


3o 


3i 


9-649781 


4-22 


9.951728 


• o5 


9.698053 


5 


27 


10.301947 


11 


32 


65oo34 


4-22 


9 5i665 


• o5 


6 9 836 9 
6 9 8685 


5 


27 


3oi63i 


33 


630287 


4-21 


931602 


-o5 


5 


26 


3oi3i5 


27 


34 


65o539 


4-21 


9D1 539I 


• o5 


699001 


5 


26 


300999 
3oo684 


26 


35 


630792 


4-21 


951476 


• o5 


699316 


5 


26 


25 


36 


65 1 044 


4-20 


951412 


• o5 


699632 


5 


26 


3oo368 


24 


12 


631297 


4-20 


95 1 349 


• 06 


699947 


5 


26 


3ooo53 


23 


65 1 349 


4-20 


951286 


.06 


700263 


5 


25 


299737 


22 


3 9 


65 1 800 


4-19 


951222 


.06 


700578 


5 


25 


299422 


21 


40 


632052 


4-19 


95n59 


.06 


700893 


5 


25 


299107 


20 


41 


7 -6323o4 


4-iQ 

4.18 


Q- 95l096 


.06 


9.701208 


5 


24 


10-298792 


» 


42 


632355 


95io32 


.06 


70i523 


5 


24 


298477 


43 


632806 


4-i8 


950968 


.06 


701837 


5 


24 


2 9 8i63 


17 


44 


633037 


4-18 


950905 


•00 


702152 


5 


24 


297848 


16 


45 


6533o8 


4-i8 


950841 


.06 


702466 


5 


24 


297534 


i5 


46 


653558 


4-17 


950778 


.06 


702780 


5 


23 


297220 


14 


47 


6538o8 


4-17 


950714 


.06 


703095 


5 


23 


296905 
296391 


i3 


48 


6340^9 


4-17 
4-16 


95o65o 


.06 


703409 
703723 


5 


23 


12 


49 


654309 


9 5o586 


.06 


5 


23 


296277 


11 


5o 


654558 


4- 16 


95o522 i 


.07 


704036 


5 


22 


295964 


10 


5i 


9-6548o8 


4-i6 


9 -95o458 


•07 


9.704350 


5 


22 


io- 295650 


9 


52 


655o58 


4-i6 


950394 i 
95o33o i 


•07 


704663 


5 


22 


295337 


8 


53 


655307 


4-i5 


•07 


704977 


5 


22 


295023 


7 


54 


655556 


4-i5 


950266 i 


.07 


705290 


5 


22 


294710 


6 


55 


6558o5 


4-i5 


950202 1 


•07 


7o56o3 


5 


21 


294397 
294084 


5 


56 


656o54 


4-i4 


9 5oi38 1 


.07 


705916 


5 


21 


4 


u 


6563o2 


4-i4 


950074 


•07 


706228 


5 


21 


293772 3 


65655i 


4-i4 


95ooio 1 


•°7 


706541 


5 


21 


293459 2 
293146J 1 


59 


6567QO 


4-i3 


949945,1 
949881 |i 


.07 


706854 


5 


21 


60 


65704 ; 


4-i3 


•07 


707166 


5-20 


292834I 




CoBino 


D. 


Sir* 163° 


Cotaiig. 


E 


►• 


"MjjTj 


M. 



8INE8 AND TANGENTS. (27 DEGREES.) 



4T, 



M. 


Sino 


D. 


Cosine j D. 


1 Tang. 


D. 


Cotang. 


L_ 





9.657047 


4-i3 


9-949881 1-07 


| 9-707166 


5-20 


10- 292834| 60 


i 


657290 


4-i3 


949816 1 -07 


707478 


5-20 


292522 


1 5 9, 


2 


657042 


4-12 


949752 i-o- 
949688 i- 08 


707790 
708102 


5- 20 


292210 


58 


3 


657790 
658o37 


4-12 


5-20 


291898 
29 1 586 


5 7 


4 


4-12 


949623 I -0* 


708414 


5- 19 


56 


5 


658284 


4-12 


949608 I -o£ 


708726 


5- 19 


291274 


55 


6 


65853i 


4- 11 


949494I i-o^ 


709037 


5-19 


290960 


54 


I 


658 77 8 


4-n 


949429' I -0^ 


! 709349 


5-19 


290661 


53 


659025 


4-n 


949364 I -o£ 


7O9660 


5- 19 

5-i8 


290340 


52 


9 


639271 


4mo 


949300 1 -o£ 


7O997I 


290029 
289718 


5 


JO 


669517 


4-io 


949235 1 -08 


710282 


5-i8 


5o 


n 


9-669763 


4-io 


9-949170 1 -08 


; 9-710693 


5- 18 


10-289407 


40 


13 


660009 
660263 


4-09 


949105 1 -oJ 


710904 


5- 18 


289096 

288786 


48 


i3 


4-09 


949040 1 -oJ 


71 1 2 I 5 


5- .8 


47 


14 


66o5oi 


4-09 


94^976; I -o£ 


71 ID25 


5- 17 


288475 


46 


i5 


660746 


4*09 


948010 1 -o£ 


7ii836 


5- 17 


288164 


45 


16 


660991 


4-o8 


948845 1 -08 


712146 


5-i 7 


287854 


44 


n 


66i236 


4-o8 


948780:1 -09 


712436 


5.17 


287544 


43 


18 


661481 


4-o8 


948715 


1 -09 


712766 


5-i6 


287234 


42 


'9 


661726 


4-07 


94865o 


1 -09 


713070 
713386 


5- 16 


286924 


4i 


20 


661970 


4-07 


948584 


1 -09 


5- 16 


286614 


40 


21 


9-662214 


4-07 


9-94831911 -09 
948434! 1 -09 


9-713696 


5-i6 


10-286004 


3 9 


22 


662459 


4-07 


714003 


5-i6 


286995 


38 


23 


66270J 


4-o6 


948388| 1 -09 


7i43i4 


5-13 


285686 


37 


24 


662946 


4-o6 


948323 1-09 


714624 


5-i5 


285376 


36 


25 


663190 


4-o6 


948267 


1 -09 


714933 


5- 15 


286067 


35 


26 


663433 


4-o5 


948192 


1 -09 


716242 


5- 13 


284758 


34 


27 


663677 


4-o5 


948126 


1 -09 


7i555i 


5-i4 


284449 


33 


28 


663920 


4-o5 


948060 


1-09 


716860 


5-i4 


284140 


32 


? 9 


664i63 


4-o5 


94799 s 


I-IO 


716168 


5-i4 


283832 


3i 


3o 


664406 


4-o4 


947929 
9-947863 


I-IO 


716477 


5-i4 


283523 


3o 


3i 


9-664648 


4- 04 


I-IO 


9-716785 


5-14 


io-2832i5 


3 


32 


664891 


4-o4 


947797 


I - 10 


717093 


5-i3 


282907 


33 


665i33 


4-o3 


94773i 


I-IO 


7'74oi 


5-i3 


282699 


27 


34 


665375 


4-o3 


947663 


I-IO 


717709 


5-i3 


282291 


26 


35 


6656 if 


4-o3 


947600 


I • 10 


718017 


5- 13 


281983 


26 


36 


66585 9 


4-02 


947533 


I-IO 


7i8323 


5-i3 


281670 


24 


ll 


666 1 00 


4-02 


947467 


I • 10 


7i8633 


5-12 


281367 


23 


666342 


4-02 


947401 


I-IO 


718940 


5-12 


281060 


22 


39 


666583 


4-02 


947335 


I-IO 


719248 


5-12 


280762 


21 


4o 


666824 


4-oi 


947269 


I-IO 


719555 


5-12 


280446 


20 


4i 


9-667066 


4-oi 


9-947203 


I-IO 


9-719862 


5-12 


io-28oi38 


8 


42 


66730D 


4-oi 


947i36 1 -ii 


720169 


5-n 


279831 


43 


667646 


4-oi 


947070 i-ii 


720476 


5- 11 


279624 


17 


44 


667786! 


4-oo 


947004 1 • 1 1 


720783 


5- 11 


279217 


16 


45 


668027: 


4-oo 


946937 
946871 


III 


721089 


5- 11 


27891 1 


i5 


46 


668267! 


4-oo 


I-II 


721396 


5. 11 


278604 


14 


s 


668 5o6! 


3'99 


946804 


I-II 


721702 


5-10 


278298 


i3 


668746 ' 


3-99 


946738! i-ii 


722009 


5- 10 


277991 


12 


* 9 


668986 


3-99 


946671 i-ii 


7223i5 


5- 10 


277686 


11 


5o 


66922D f 


3-99 


946604)1 -u 


722621 


5-io 


277379 
10^277073 


10 


5i 


9.669464: 


3.98 


9 946538 


I-II 


9.722927 


5»io 


9 


52 


669703 


3.98 


94647 1 


I-II 


723232 


5-09 


276768 


8 


53 


669942 


3- 9 8 


946404 


I-II 


723538 


5-09 


276462 


7 


54 


670181] 


3-97 


946337 


I-II 


723844 


5-09 


2-6i56l 6 


55 


670419 
670658 


3-97 


946270 


I- 12 


724149 


5-09 


27585i 


5 


56 


3-97 


946203 


I -12 


724454 


5-09 
5-o8 


275546 


4 


tl 


670896 
671 1 34 


3-97 


946 1 36 


I- 12 


724759 


276241 


3 


3- 9 6 


946069 


I-I2I 


726066 


5.o8 


274935 


2 


^ 


671372 


3- 9 6 


946002! 1 • 12 


725369 


5-o8 


274631 


1 


60 


671609 


3- 9 6 


945935 I -12 


725674 


5- 08 


274326I 


1 


Cosine ! 


1>. 


Sine ^62° 


Cotang. 


D Tang. 1 


M. 



46 



(28 DEGREES.) A TABLE OP LOGARITHMIC 



M. 


Sino 


D. 


Cosmo | D. 


Tang. 


1 D. 


Cotang. 


._- 


o 


9-671609 


3.96 


9-945o35 1. 13 


9.725674 


5-o8 


10-274326 


60" 


i 


671847 


3. 9 5 


945868 1. 1 2 


725979 
726284 


I 5- 08 


274021 


5 5 


9 


672084 


3. 9 5 


945800 i- 1 2 


' 5-07 


273716 


58 


3 


672321 


3. 9 5 


945733 i- 1 2 


72658S 


| 5#0 7 


273412 


5 7 


4 


672558 


3. 9 5 


945666 1 • 1 2 


726892 


5-07 


273108 


56 


5 


672795 
673032 


3-94 


945598 I -12 


727197 


5 -07 


272803 


55 


6 


3.94 


94553i 1. 1 2 


727501 


! 5.07 


272499 54 


I 


673268 


3.94 


945464 I-K 


727805 


5.06 


272195 


53 


67350D 


3.94 


945396 I • K 


728109 


1 5. 06 


271891 
271588 


52 


9 


6 7 3 7 4i 


3. 9 3 


945328 i - 13 


728412 


: 5-o6 


5i 


10 


673977 


3. 9 3 


94526l I-K 


728716 


5.o6 


271284 


5o 


ii 


9-674213 


3- 9 3 


9-945I93 I-K 


9.729020 


5- 06 


10-270980 


49 


12 


674448 


3-92 


945i25 i-k 


729323 


! 5-o5 


270677 


48 


i3 


674684 


3-92 


945o58 no 


729626 


' 5-o5 


270374 


47 


14 


674919 


3-92 


944990 1. 1 3 


729929 
73o233 


1 5-o5 


270071 


46 


i5 


6 7 5i55 


3-92 


944922 1 - 13 
944854 1- 1 3 


5-o5 


269767 


45 


16 


675390 


3-91 


73o535 


5-o5 


269465 


44 


\l 


675624 


3-91 


944786 i- 1 3 


73o838 


5-o4 


269162 


43 


675859 


3gi 


944718 1 -i3 


731141 


5-04 


26885 9 


41 


»9 


676094 


3-91 


94465o 1 . i3 


73i444 


i 5-04 


268556 


4i 


20 


676328 


3-90 


944582 1 • 14 


73i746 


5-04 


268254 


40 


21 


9-676562 


3 90 


9-9445i4 1 -14 


9-732048 


5 /°i 


10-267952 


3 9 


22 


676796 
6770.50 


3.90 


944446 1-14 


73235i 


5-o3 


267649 


38 


23 


3.90 


944377 1 -14 


732653 


5-o3 


267347 


37 


24 


677264 


3.^9 


944309 1. 14 


732955 


5-o3 


267045 


36 


25 


677498 


3-89 


944241 1 1 -14 


733257 


5-o3 


266743 


35 


26 


677731 


3-89 
3.88 


944H 2 1 • 14 


733558 


5-o3 


266442 


34 


*7 


677964 


944104 1 -14 


73386o 


5-02 


266140 


33 


jS 


678197 


3-88 


944o3 6 1 -14 


734162 


5-02 


265838 


32 


29 


6?843o 


3-88 


943967 1.14 


734463 


5-02 


26553 7 


3i 


3o 


678663 


3-88 


943899 1 .14 
9-94383o;i -14 


734764 


5-02 


265236 


3o 


3i 


9-678895 


3-8 7 


9- 735o66 


5-02 


10-264934 


29 


32 


679128 


3-8 7 


943761 ji. 14 


735367 


5-02 


264633 


28 


33 


679360 


3-8 7 


943693 1 - 1 5 


735668 


5-oi 


264332 


27 


34 


679092 


3-8 7 


943624 1 - 15 


735969 


5-oi 


264031 


26 


35 


679824 


3-86 


943555 1 - 15 


736269 


5-oi 


263731 


25 


36 


68oo56 


3-86 


943486 1. 1 5 


736570 


5-oi 


26343o 


24 


37 


680288 


3-86 


943417 1 • 1 5 


736871 


5-oj 


263129 


23 


38 


680D19 


3-85 


943348 1 - 15 


737171 


5-oo 


262829 


22 


. 3 9 


680750 


3-85 


943279 1 -i5 


737471 


5-oo 


262529 


21 


4o 


680982 


3-85 


9432io 1 - 1 5 


737771 


5-oo 


262229 


20 


4i 


9-68i2i3 


3-85 


9-943i4i 1 • i5 


9-738071 


5-oo 


10-261929 


:? 


42 


68i443 


3.84 


943072 1 • 1 5 


738371 


5- 00 


261629 


43 


681674 


3-84 


943oo3 1 ■ i5 


738671 


4-99 


261329 


17 


44 


68i 9 o5 


3-84 


942934 1 • 1 5 


738971 


4.99 


261029 


16 


45 


682135 


3 -84 


942864 1 - 15 


739271 


4-99 


260729 1 5 


46 


682365 


3-83 


942795 i- 16 


739570 


4.99 


26o43o 14 


4 I 


6825 9 5 


3-83 


942726 i- 16 


739870 


4.99 


26oi3o i3 


48 


682825 


3-83 


942656 1 - 16 


740169 
740468 


4.99 


25983ij 12 


49 


683o55 


3-83 


942587|i-i6 


4.98 


259532 11 


5o 


683284 


3-82 


942517 1 • 16 


740767 


4-98 1 


» s5o233 10 

10-258934 9 

258635, 8 


5i 


9-6835i4 


3-82 


9-942448 i- 16 


9.741066 


4-98 ; 


52 


683743 


3.82 


942378.1-16 


74i365 


4- 9 8 


53 


683972 


3-82 


9423o8!i-i6 


741664 


4-98 


258336' 7 


54 


684201 


3-8i 


942239'i-i6 


741962 


4-97 


258o38 6 


55 


68$43o 


3.8i 


942169 i- 16 


742261 


4-97 


257739! 5 


56 


6846581 


3-8i 


942099 1 - 16 


742559 

742858 


4-97 


257441' 4 


n 


684887 | 


3-8o 


942029 1 -16 


4-97 


257142: 3 


685n5j 


3.8o 


941959 1 - 16 


743 1 56 


4-97 


256844' 2 


5q 


685343! 


3-8o 


941889 1-17 


743454 


4-97 
4-96 
D. 


256546! 1 


6o 


6855 7 1 | 


3-8o 


941819 1-17 


743752 


256248! 


1 


Cosine 1 


D. 


Sino Gl° 


Cctang. 


^an^T 


KL 





SINES AND TANGENTS. 


(29 DEGREES. 


) 


47 


M. 


Sine 


D. 


Cosine | D. 


Tang. 


1 D- 


Cotang. | 





9-685571 


3-8o 


9-941819 


1-17 


9-743752 


4 


96 


10-256248 60 


i 


685799 


a- 79 


941749 


117 


744o5o 


4 


96 


255 9 5o 5o 
255652 58 


2 


686027 


3-79 


941679 


1-17 


744348 


4 


96 


3 


686254 


3-79 


941609 


117 


744645 


4 


96 


255355 5 7 


4 


686482 


£:? 


94i539 


1-1-7 


744943 


4 


96 


255o57| 56 


5 


686709 


941469 


1-17 


745240 


4 


96 


2547601 55 


6 


686 9 36 


3- 7 8 


94 1 398 


1-17 


745538 


4 


9 5 


2 04462 1 54 


I 


687163 


3- 7 8 


94i328 


117 


745835 


4 


95 


254i65 53 


687389 


3.78 


941258 


1-17 


746i32 


4 


95 


253868 


52 


9 


687616 


3-77 


94i 187 


1-17 


746429 


4 


95 


253571 


5i 


IO 


687843 


3-77 


94i 1 17 


I- 17 


746726 


4 


95 


253274 


5o 


ii 


9-688069 


3-77 


9 • 94 1 046 


1 - 18 


9-747023 


4 


94 


10-252977 


A l 


12 


68829D 


3.77 


940Q75 


1. 18 


7473i9 


4 


94 


25268i 


48 


i3 


6&8021 


3-76 


940005 


1-18 


747616 


4 


94 


252384 


47 


i4 


688747 


3.76 


940834 


i- 18 


747913 


4 


94 


202087 


46 


i5 


688972 


3.76 


040763 


1- 18 


748209 


4 


94 


201791 


45 


16. 


689198 


3.76 


940693 


1 - 18 


7485o5 


4 


93 


25i495 


44 


17 


689423 


3.75 


940622 


1. 18 


748801 


4 


93 


25i 199 


43 


1 8 


689648 


3- 7 5 


94o55i 


1-18 


749097 


4 


93 


25o9o3 


42 


19 


689873 


3- 7 5 


940480 


1 • 18 


7493o3 


4 


9 3 


250607 


4i 


20 


690098 


3- 7 5 


940409 


1- 18 


749689 


4 


93 


25o3u 


40 


21 


9 690323 


3-74 


9-94o338 


1. 18 


9-7499 8 5 


4 


93 


iO'25ooi5 


u 


22 


690548 


3.74 


940267 


1 -18 


750281 


4 


92 


249719 


23 


690772 


3-74 


940 1 96 


1. 18 


750576 


4 


92 


249424 


37 


24 


690996 


3 74 


940120 


1. 19 


750872 


4 


92 


249128 
248833 


36 


25 


691220 


3.73 


940054 


1-19 


751167 


4 


92 


35 


26 


691444 


3- 7 3 


939982 


1-19 


701462 


4 


92 


248538 


34 


27 


691668 


3- 7 3 


93991 1 


i- 19 


751757 


4 


92 


248243 


33 


28 


691892 


3.73 


Q 3 9 84o 


1-19 


752o52 


4 


9' 


247948 


32 


29 


6921 1 5 


3.72 


939768 


1-19 


752347 


4 


9i 


247653 


3i 


3o 


692339 


3. 7 2 


939697 


1-19 


752642 


4 


9' 


247358 


3o 


3i 


9-692562 


3.72 


9-939625 


1-19 


9-752937 


4 


9i 


10-247063 


2 


32 


692780 


3.71 


939554 


I- 19 


75323i 


4 


9i 


246769 


33 


693008 


3.71 


939482 


1-19 


753526 


4 


9' 


246474 


27 


34 


693231 


3.71 


939410 


1-19 


753820 


4 


9° 


246180 


20 


35 


693453 


3.71 


939339 


I 19 


7541 i5 


4 


90 


245885 


25 


36 


693676 


3-70 


939267 


I- 20 


754409 


4 


90 


245591 


24 


37 


693898 


3-70 


Q39195 


1 -20 


754703 


4 


90 


245297 


23 


38 


694120 


3-70 


939123 


I- 20 


754997 


4 


90 


245oo3 


22 


39 


694342 


3-70 


93oo52 


1-20 


755291 


4 


5 


244709 
2444i5 


21 


40 


694564 


3.69 


938980 


1-20 


755585 


4 


20 


41 


9-694786 


3-6g 


9.938908 

938836 


1-20 


9 -755S 7 8 


4 


*9 


10-244122 


18 


42 


695007 


3.69 


1-20 


756172 


4 


89 


243828 


43 


695229 


3-6 9 
3-68 


938763 


I- 20 


756465 


4 


8 9 


243535 


17 


44 


695450 


938691 


I '20 


756759 


4 


89 


243241 


16 


45 


695671 


3-68 


938619 


1-20 


757052 


4 


a 


242948 


i5 


46 


695892 


3-68 


9 38547 


1-20 


757345 


4 


242655 


14 


41 


696 1 1 3 


3-68 


938475 


1-20 


757638 


4 


88 


242362 


i3 


48 


696334 


3-6 7 


938402 


I -21 


757931 


4 


88 


242069 


12 


49 


696554 


3.67 


93833o 


I -21 


758224 


4 


88 


241776 


11 


5o 


696775 


3.67 


938258 


I - 21 


7585i7 


4 


88. 


24U83 


10 


5i 


9-696995 


3.67 


9-938i85 


I -21 


9-758Sio 


4- 


88 


IO-24HQO 





52 


697215 


3-66 


9381 i3 


I -21 


759102 


4- 


87 


240898 


8 


53 


697435 


3-66 


938040 


I • 21 


759395 


4- 


87 


24o6o5 


7 


54 


697654 


3-66 


937967 


I -21 


759687 


4- 


87 


24o3i3 


6 


55 


697874 


3-66 


937895 


I-2I 


709979 


4- 


87 


240021 


5 


56 


698094 


3-65 


937822 


I-2I 


760272 


4- 


87 


239728 


4 


57 


6 9 83 1 3 


3-65 


937749 


I -21 


760564 


4- 


87 


239436 


3 


58 


698532 


3-65 


937676 


I-2I 


76o856 


4- 


86 


239144 


2 


5 9 


698751 


3-65 


937604 


I-2I 


761 148 


4- 


86 


238852 


1 


60 


698970 


3-64 


937531 


I-2I 


761439 


4- 


86 


23856i 
Tang. 





Cosine 


D. 


Sine 


GOO 


Cotang. 


_J 


X 



18 


(30 


DEGREES.) A TABLE OP LOQAKITHMIC 




M. 


Sine 


IX 


Cosine | D. 


Tang. 


D. 


Cotang. 


60 





9-698970 


3 


64 


9*93753i'i >2i 


9-761439 


4-86 


io-23856i 


i 


699189 


3 


• 64 


937458 1 


•22 


761731 


4-86 


238269 


n 


2 


699407 


3 


-64 


9 37385 1 


•22 


762023 


4-86 


237977 


3 


699626 


3 


64 


937312^ 


•22 


762314 


4-86 


237685 


57 


4 


699844 


3 


67 


937238 1 


•22 


762606 


4-85 


237394 


56 


5 


700062 


3 


oJ 


937165 1 


•22 


762807 


4-85 


237103 


55 


6 


700280 


3 


63 


937092 1 


•22 


763i88 


4-85 


236812 


54 


I 


700498 


3 


63 


937019 1 


•22 


7634-9 


4-85 


236521 


53 


700716 


3 


63 


936946 1 


•22 


763770 


4-85 


2362 3o 


52 


9 


700933 


3 


62 


936872 1 


•22 


764061 


4-85 


235939 5i 


IO 


701 :5i 


3 


62 


936799 1 
9-936725 1 


•22 


764352 


4-84 


235648 


5o 


ii 


9-70i368 


3 


62 


•22 


9-764643 


4.84 


i3-23535 7 


% 


12 


7oi585 


3 


62 


936652 1 


•23 


764933 


4-84 


235067 


i3 


701802 


3 


6i 


936578 1 


•23 


765224 


4-84 


234776 


47 


14 


702019 


3 


61 


9365o5 1 


•23 


7655i4 


4-84 


234486 


46 


15 


702236 


3 


61 


93643i 1 


•23 


7658o5 


4-84 


234195 


45 


16 


702432 


3 


61 


936357 1 


•23 


766095 


4-84 


233905 
2336 1 5 


44 


\l 


702669 
702885 


3 


60 


936284 1 


•23 


766385 


4-83 


43 


3 


60 


936210 1 


•23 


766675 


4-83 


233325J 42 


*9 


7o3ioi 


3 


60 


936 1 36 1 


•23 


766965 


4-83 


233o35| 41 


20 


703317 


3 


60 


936062 1 


•23 


767255 


4-83 


232745' 40 


21 


9 -7o3533 


3 


5 9 


9 -935988 1 


•23 


9-767545 


4-83 


10-232455 39 
232i66 : 38 


22 


703749 


3 


5 9 


935qi4!i 
935840 1 


•23 


767834 


4-83 


23 


703964 


3 


5 9 


•23 


768124 


4-82 


231876; 37 


24 


704179 
704395 


3 


5 9 


935766 1 


■24 


768413 


4-82 


23i587i 36 


25 


3 


tl 


035692(1 


■24 


768703 


4-82 


2312971 35 


20 


704610 


3 


9356i8!i 


• 24 


768992 


4-82 


23 1 008: 34 


3 


704825 


3 


58 


935543[i 


•24 


769281 


4-82 


230719 1 33 


7o5o4o 


3 


58 


935469^1 


•24 


769570 


4-82 


23o43o 32 


20 


703204 


3 


58 


935390 1 


• 24 


769860 


4-8i 


23oi4o! 3i 


3o 


705469 


3 


57 


935320 1 


• 24 


770148 


4-8i 


22Q852J 3o 


3i 


9«7o5683 


3 


57 


9-935246 1 


•24 


9.770437 


4-81 


io. 2 2 9 563l 29 


32 


705898 


3 


57 


935171 1 


•24 


770726 


4-8i 


229274 


28 


33 


7061 12 


3 


57 


935097 1 


•24 


771015 


4-81 


228985 


27 


34 


706326 


3 


56 


935022 1 


•24 


77i3o3 


4-8. 


228697 


26 


35 


706539 


3 


56 


934948 1 


•24 


771592 


4-8i 


228408 


25 


36 


706753 


3 


56 


934873 1 


•24 


771880 


4-8o 


228l 20' 24 


s 


706967 


3 


56 


934798; 1 


•25 


772168 


4-8o 


227832 


23 


707180 


3 


55 


934723|i 


•25 


772457 


4-8o 


227543 


22 


3 9 


707393 


3 


55 


934649 1 


•25 


772745 


4-8o 


227255 


21 


4o 


707606 


3 


55 


934574' 1 


•25 


773o33 


4-8o 


226967 


20 


41 


9.707819 


3 


55 


9-934499; ] 


•25 


9-773321 


4-8o 


'O.226679 


\l 


42 


708032 


3 


54 


934424(1 


•25 


773608 


4-79 


226392 


43 


708245 


3 


54 


934349;' 


•25 


773896 


4-79 


226104 


'7 


44 


708458 


3 


54 


934274,1 


•25 


774184 


4-79 


2258i6 16 


45 


708670 


3 


54 


934!99 !l 


•25 


774471 


4-79 


225529 i5 


46 


708882 


3 


53 


9341 23 ( I 


•25 


774759 


4-79 


225241 


14 


47 


709094 


3 


53 


934048 1 


•25 


775046 


4-79 


224954 


i3 


48 


709306 


3 


53 


933973 1 


•25 


775333 


4-79 
4-78 


224667 


12 


49 


709518 


3 


53 


933M(i 


• 26 


775621 


224379 


11 


5o 


709730 


3 


53 


933822 1 ! 


• 26 


775908 


4-78 


224092 


10 


5i 


9-709941 


3 


52 


9-933747 1 


• 26 


9.776195 


4-78 


«o 2238o5 


% 


52 


7ioi53 


3 


52 


933671 1 


• 26 


776482 


4-78 


2235i8 


53 


7io364 


3 


52 


933596' 1 


■ 26 


776769 
777055 


4-78 


22323l 


7 


54 


710075 


3 


52 


93352oi 1 


• 26 


4-78 


222945 


6 


55 


710786 


3 


5i 


933445 1 


• 26 


777342 


4 78 


222658 5 


56 


710997 


3 


5i 


933369 1 


• 26 


777628 


4-77 


222372! 4 


u 


711208 


3 


5i 


933293 1 


• 26 


777915 


4-77 


222085j 3 


7*1419 


3 


5i 


933217 1 


• 26 


778201 


4-77 


221799; 2 


59 


711629 


3 


5o 


933i4i 1 


• 26 


778487 


4-77 


2 2 I 5 1 2 i I 


6o 


71 1839 


3 


5o 


933o66 1 


• 26 


778774 


4-77 


221226, 


Cosine 


D. 


Sine £ 


9° 


Cotaiig. 


D. 


Taiig._ 


M.J 





SINES AND TANGENTS. 


(31 DEGREES. 


) 


4 





Sine 


D. 


| Cosine | D. 


| Tang. 


D. 


Cotang. 




9-711839 


3-5o 


1 9-933o66|i-26| 9-778774 


4-77 


10-221226 


~6o~ 


2 


7i2o5o 
712260 


3-5o 
3-5o 


932990 1-27 
9 32oi4ii-27 
9 32838{i-27 


j 779060 
779346 


4-77 
4-76 


220940 
220654 


5c 
58 


3 


712469 


3-49 


779632 


4-76 


220368 


57 


4 


712679 


3-49 


9327621 -27 


779918 


4-76 


220082 


56 


5 


712889 
713098 


3-49 


q32685| 1 - 27 


780203 


4-76 


219797 


55 


6 


3-49 


932609 1 -27 


780489 


4-76 


2 1901 1 , 54 


I 


7i33o8 


3-4C 
3-48 


9325331-27 


1 780775 


4-76 


2IQ225! 53 


7i35 1 7 


9324571-27 


, 781060 


4-76 


218940; 52 


9 


713726 


3-48 


93238o 1 -27 


781346 


4-75 


2i8654| 5i 


IO 


713935 


3-48 


9323o4 1 -27 


78i63i 


' 4-75 


2183691 5o 


ii 


9-714144 


3-48 


9 932228 1-27 


9-781916 


; 4-73 


10 218084 49 


12 


714352 


3-47 


9321 5i 1 -27 


782201 


4-7 = 


2H799 


48 


i3 


7i456i 


3-47 


932075 1 -28 


782486 


4-75 


217514 


47 


14 


714769 
714978 


3-47 


931998,1 -28 


782771 


4-75 


217229 


46 


i5 


3-47 


931921 ji -28 


783o56 


4-75 


216944 


45 


16 


7 1 5 i 86 


3-47 


93i845 1-28 


783341 


4-75 


216659 


44 


17 


715394 


3-46 


931768 1-28 


783626 


4-74 


216374 


43 


18 


715602 


3-46 


93 1691 1-28 


783910 


4-74 


216090 


42 


19 


7 1 5809 


3-46 


931614 1-28 


784195 


4-74 


2i58o5 


4i 


20 


716017 


3-46 


93 1 537 1 -28 


784479 


4-74 


2I552I 


40 


21 


9-716224 


3-45 


9-931460 1 -28 


9-7B4764 


4-74 


io-2i5236 


ll 


22 


716432 


3-45 


931383^-28 


785048 


4-74 


214952 


23 


716639 


3-45 


93i3o6 1 -28 


7 85332 


4-73 


214668 


u 


24 


716846 


3-45 


93i 229J1 • 29 


7856i6 


4-73 


214384 


25 


717053 


3-45 


93ii52 1 -29 


785900 


4-73 


214100 


35 


26 


717259 


3-44 


931075,1 -29 


786184 


4-73 


2i38i6 


34 


a 


7H466 


3-44 


930998 1-29 


786468 


4-73 


2i3532 


33 


717673 


3-44 


93092 j 1 -29 
93o843 1-29 


786752 


4- 7 3 


213248 


32 


29 


717379 


3-44 


787036 


4- 7 3 


2 1 2964 


3i 


3o. 


7i8o85 


3-43 


930766 1 -29 


787319 


4-72 


212681 


3o 


3i 


9-718291 


3-43 


9.9306881-29 


9-787603 


4-72 


10-212397 


3 


32 


718497 


3-43 


930611 1 -29 


7878S6 


4-72 


2121 14 


33 


718703 


3-43 


9 3o533 1.29 


788170 


4-72 


2ii83o 


27 


34 


718909 


3.43 


93o456 1 -29 


788453 


4-72 


2 I I D47 


26 


35 


7i9"4 


3-42 


93o378 1-29 


7 88 7 36 


4-72 


21 1264 


25 


36 


719320 


3-42 


93o3oo 1 -3o 


789019 


4-72 


2 1 098 1 


24 


37 


719525 


3-42 


93o223 1 -3o 


789302 


4-71 


2 1 0698 


23 


38 


719730 


3-42 


93oi45 i-3o 


7S 9 585 


4-71 


2 1 04 1 5 


22 


3 9 


719935 


3-41 


930067 1 -3o 


789868 


4-71 


210132 


21 


4o 


720140 


3-4i 


929980 1 -3o 


7901 5 1 


4-71 


209849 


20 


4i 


9-72o345 


3-4i 


9-92991 1 ■ 1 -3o 
929833 1 -3o 


9-790433 


4-71 


IO-209567 


\i 


42 


720549 


3-4i 


790716 


4-71 


209284 


43 


720754 


3-40 


929755^ -3o 


790999 
79 1 28 1 


4-7i 


209OOI 


H 


44 


720958 


3-4c 


929677,1.30 


4-71 


208719 


16 


45 


721162 


3-40 


929D99 i-3o 


79 1 563 


4-70 


208437 


i5 


46 


72i366 


3-4o 


929D21 1 -3o 


791846 


4-7o 


208 1 54 


14 


47 


72157c. 


3.40 


929442 1 -3o 


792128 


4-70 


207872 


i3 


48 


721774 


3-3 9 


929364 1 -3i 


792410 


4-70 


207 D90 


12 


i 9 


721978 


3.39 


929286 1 - 3 1 


792692 


4-70 


2 ">73o8 


11 


5o 


722181 


3-39 


929207 1 -3i 


792974 


4-70 


207026 


10 


5i 


9-722385 


3-3 9 


9-9291291 -3i 


9-793256 


4-70 


10-206744 


i 


52 


722588 


3.39 
3-38 


92905c 1 -3 1 
928972 i-3i 


79 3538 


4-6 9 


206462 


53 


722791 


793819 


4-6 9 


206l8l 


I 


54 


722994 


3-38 


928893 1 -3 1 


794ioi 


4-6 9 


205899 


55 


723197 


3-38 


928815, i-3i 


794383 


4-6 9 


205617 


5 


56 


723400 


3-38 


928736' 1-3 I! 


794664 


4-6 9 


205336 


4 


Si 


7236o3 


3.37 


928637 1. 3 1 


794945 


4-6 9 


2o5o55 


3 


7238o5| 


3-3 7 


928378,1 -3i ! 


795227 


4-69 
4-68 


204773 


2 


5o | 


724007 


3.37 


928499 i-3i 


7955o8 


204492 


1 


6o 1 


724210 


3.37 ! 


928420J1 -3i 

Sine |58° 


795789 


4-68 


20421 1 





Cosine 


D. 


Cotang. 


D. | 


Tang. 


M. 



50 


(32 DEGREES.) A TABLE OF LOGARITHMIC 




M. 
o 


Sine 


D. 


Cosine | T>. 


Tang. 


D. 


Cotang. 




9-724210 


3.37 


9-928420 1-32 


9 -795789 


4-68 


10-204211 


60 


i 


724412 


3-37 


928342 1 


■32 


796070 


4-68 


203930 


u 


2 


724614 


3-36 


928263 1 


•32 


796351 


4-68 


203649 


3 


724816 


3-36 


9 28i83ji 


•32 


796632 


4-68 


203368 


u 


4 


725017 


3-36 


928104 1 


•32 


796913 


4-68 


203087 


5 


72D219 


3-36 


928o25'i 


•32 


797I94 


4-68 


202806 


55 


6 


723420 


3-35 


927946 1 


•32 


797475 


4-68 


202525 


54 


2 


723622 


3-35 


927867 1 


•32 


797755 


4-68 


202245 


53 


725823 


3-35 


92778-7 1 


•32 


798036 


4-67 


201964 


52 


9 


726024 


3-35 


927708 1 


■32 


7 9 83 1 6 


4-67 


201-684 


5i 


10 


726225 


3-35 


927629 1 


•32 


798596 


4-67 


201404 


5o 


ii 


9-726426 


3-34 


0-927549 1 


•32 


9.798877 


4-67 


IO-20I 123 


% 


12 


726626 


3-34 


927470 1 


•33 


799157 


4-67 


200843 


i3 


726827 


3-34 


927390 1 


• 33 


799437 


4-67 


2oo563 


47 


14 


727027 


3-34 


927310 1 


■33 


7997H 


4-67 


200283 


46 


i5 


727228 


3-34 


927231 1 


•33 


799997 


4-66 


200003 


45 


16 


727428 


3-33 


927 1 5i ; 1 


•33 


800277 


4-66 


199723 


44 


\l 


727628 


3-33 


927071 ji 


•33 


8oo557 


4-66 


199443 


43 


727828 


3-33 


926991 1 


•33 


8oo836 


4-66 


199164 


42 


19 


728027 


3-33 


9269I ! I 


•33 


801 1 16 


4-66 


I9S884 


4i 


20 


728227 


3-33 


92683i Ii 


•33 


801396 


4-66 


198604! 40 


21 


9-728427 


3-32 


9-926701 j 1 


•33 


9-801675 


4-66 


io- 198325 39 
198045 38 


22 


7 2 86 26 


3-32 


o2667i'i 


•33 


8 [ 9 55 


4-66 


23 


728825 


3-32 


926591 j 1 


•33 


802234 


4-65 


197766! 37 


24 


729024 


3-32 


9265 1 11 


•34 


8o25i3 


4-65 


197487J 36 


25 


729223 


3-3i 


92643i|i 


■34 


802792 


4-65 


197208 35 


26 


729422 


3-3i 


92635i 1 


■34 


803072 


4-65 


196928, 34 
196649 33 


2 I 


■72Q621 


3-3i 


926270! 1 


•34 


8 335i 


4-65 


28 


729820 


3-3i 


926190 1 


•34 


8o363o 


4-65 


196370 32 


29 


73ooi8 


3-3o 


9261 ioj 1 


•34 


803908 


4-65 


196092 


3i 


3o 


730216 


3-3o 


926029:1 


•34 


804187 


4-65 


i 9 58i3 


. 3o 


3i 


9-73o4i5 


3-3o 


9-925949 1 
925868 1 


■34 


9 • 804466 


4-64 


io- 195534 


3 


32 


73o6i3 


3-3o 


•34 


8o4745 


4-64 


195255 


33 


73o8i 1 


3-3o 


925788 1 


•34 


8o5o23 


4-64 


194977 
194698 


27 


34 


731009 


3-29 


925707)1 


•34 


8o53o2 


4-64 


26 


35 


73 i 206 


3-29 


920626' 1 


•34 


8o558o 


4-64 


194420 


25 


36 


731404 


3-29 


925545 1 


•35 


8o585 9 


4-64 


194141 


24 


12 


731602 


3-29 


925465 1 


• 35 


8o6i3 7 


4-64 


i 9 3863 


23 


731799 


3-29 

3-28 


92538411 


■ 35 


8064 1 5 


4-63 


193585 


22 


3 9 


731996 


92o3o3 1 


•35 


806693 


4-63 


193307 


21 


4o 


732193 


3-28 


925222 1 


• 35 


806971 


4-63 


193029 


20 


4i 


9-732390 
73258 7 


3-28 


9-925141 1 


•35 


9-807249 


4-63 


10-192751 


a 


42 


3-28 


92^060 1 


• 35 


807527 


4-63 


• 192473 


43 


732784 


3-28 


924979 ' 


• 35 


807805 


4-63 


192195 


12 


44 


732980 


3-27 


924897 1 


•35 


8o8o83 


4-63 


191917 


45 


733177 


3-27 


924816 1 


-35 


8o836i 


4-63 


191639 


:5 


46 


733373 


3.27 


924735 1 


■36 


8o8638 


4-62 


191362 


14 


47 


733569 


3. 2? 


924654 1 


•36 


808916 


4-62 


191084 


i3 


48 


7 33 7 65 


3-27 


924572 1 


•36 


809193 


4-62 


190807 


12 


49 


733961 


3-26 


924491 ! 1 


•36 


809471 


4-62 


190529 


11 


5o 


734157 


3 • 26 


924409 1 


■36 


809748 


4-62 


190252 


10 


5i 


9-734353 


3-26 


9-924328:1 


-36 


9-8ioo25 


4-62 


10-189975 


8 


52 


734549 


3-26 


924246 1 


•36 


8io3o2 


4-62 


189698 


53 


734744 


3-25 


924164 1 1 


•36 


8io58o 


4-62 


189420 


1 


54 


734939 


3-25 


924083 1 


-36! 


8io85 7 


4-62 


189143 
188866 


6 


55 


735 1 35 


3-25 


924001 1 


•36 


8iii34 


4-6i 


5 


56 


73533o 


3-25 


923919 1 


• 36i 


81 1410 


4-61 


i885 9 o 


4 


12 


735525 


3-25 


923837 1 


• 36! 


81 1687 


4-6i 


i883i3 


3 


735719 


3-24 


923755 j 


• 3 7| 
•37, 


81 1964 


4-6i 


i88o36 


2 


59 


735914 


3-24 


923673 1 


812241 


4-61 


187739 1 


6o 


736109 


3-24 


923591 j 1 


37; 


812517 


4-6i 


1 874831 





Cosine 


D. 


Sine |5T°i 


Cotang. 


D. 


Tang. 1 


M. 





6INEE 


AND TANGENTS. 


(33 DEGREES. 


) 


51 


[■* 


bine 


D. 


Cosine 


D. 


Tang. 


D. 


Cotang. 







9-736109 
7363o3 


3-24 


9.923591 


t -3 7 


9-8i25i7 


4-6i 


10-187482 


60 


I 


3 


24 


923509 


i-3 7 


812794 


4-6i 


187206 


3 


2 


736498 


3 


24 


923427! 


t -3 7 


813070 


4-6i 


1 86930 


3 


736692 


3 


23 


923345! 


[.3 7 


8i3347 


4-6o 


186653 


57 


4 


736886 


3 


23 


923263 


1-37 


8i3623 


4-6o 


186377 


5b 


5 


737080 


3 


23 


923181 ; 


•37 


8 1 38 99 


4-6o 


1 86101 


55 


6 


737274 


3 


23 


923098 


-3 7 


814175 


4-6o 


185825 


54 


5 


737467 


3 


23 


923oi6j 


-3 7 


8i4452 


4-6o 


185548 


53 


737661 


3 


22 


922933I 


-3 7 


814728 


4-6o 


l852 7 2 


52 


9 


737855 


3 


22 


922851' 


•3 7 


810004 


4-6o 


184996 


5i 


IC 


738048 


3 


22 


922768. 


-38 


815279 
9 .8i5555 


4-6o 


184721 


5o 


II 


9*738241 


3 


22 


9-9226861 


-38 


4-5g 


10-184445' 49 


n 


738434 


3 


22 


922603, 


-38 


8i583i 


4-5 9 


1 841 69 


48 


:3 


738627 


8 


21 


922D20 1 


-38 


816 107 


4-5 9 


i838 9 3 


47 


14 


738820 


3 


21 


922438 


-38 


8i6382 


4-5 9 


i836i8 


46 


i5 


739013 


3 


21 


922355 


-38 


8i6658 


4-5 9 


183342 


45 


16 


739206 


3 


21 


922272 


-38 


8i6 9 33 


4-5 9 


183067 


44 


17 


73 9 3 9 8 


3 


21 


922189 


-38 


817209 


4-5 9 


182791 


43 


18 


739590 


3 


20 


922106 


-38 


817484 


4-5 9 


i825i6 


42 


»9 


739783 


3 


20 


922023 


-38 


817759 
8i8o35 


4-5o 


182241 


4i 


20 


739070 


3 


20 


921940 
9-921857 


-38 


4-58 


181965 


40 


21 


9-74oi67 


3 


20 


-3 9 


9-8i83io 


4-58 


10-181690 


ll 


22 


74o359 


3 


20 


921774 


.39 


8i8585 


4-58 


i8i4i5 


23 


74o55o 


3 


•9 


921691 


• 39 


818860 


4-58 


181140 


37 


24 


740742 


3 


J 9 


921607 


• 39 


8i 9 i35 


4-58 


i8o865 


36 


2D 


740934 


3 


19 


921524 


• 39 


819410 
819684 


'4-58 


180590 


35 


26 


741 125 


3 


19 


921441 


• 39 


4-58 


i8o3i6 


34 


27 


74i3i6 


3 




921357 1 


.39 


819959 


4-58 


1 8004 1 


33 


28 


741 5o8 


3 


l8 


921274 1 


• 39 


820234 


4-58 


179766 


32 


29 


741699 


3 


l8 


921190 1 


• 39 


82o5o8 


4-57 


179492 


3i 


3o 


741889 


3 


l8 


92 1 1 07 1 


• 3o 


820783 


4-57 


179217 
10-178943 


3o 


3i 


9-742080 


3 


18 


9-921023 1 


•3 9 


9-821057 


4-57 


a 


32 


742271 


3 


18 


920939 1 
920856 1 


• 40 


82i332 


4-57 


178668 


33 


742462 


3 


17 


• 40 


821606 


4.57 


178394 


3 


34 


742652 


3 


17 


920772 I 


• 40 


821880 


4-57 


178120 


35 


742842 


3 


17 


920688I1 


.40 


822154 


4-5 7 


177846 


25 


36 


743o33 


3 


•7 


920604 1 


• 40 


822429 
82270J 


4-57 


177571 


24 


37 


743223 


3 


17 


920520J1 


.40 


4-57 


177297 


23 


38 


7434 1 3 


3 


16 


92o436|i 


• 40 


822077 


4-56 


177023 


22 


39 


7436o2 


3 


16 


920352 1 ! 


.40 


82325o 


4-56 


176750 


21 


I 4o 


743702 


3 


16 


920268I1 


• 40 


823524 


4-56 


176476 


20 


4i 


9-7^982 


3 


16 


9-920184 I 


.40 


9-823798 


4-56 


io- 176202 


» 


42 


744i 7 1 


3 


16 


920090.1 


• 40 


824072 


4-56 


175928 


43 


74436i 


3 


i5 


920015 I 


.40 


824345 


4-56 


175655 


17 


44 


74455o 


3 


i5 


9 1 9Q3 1 J 1 

919846! 1 


•41 


824619 


4-56 


I7538i 


16 


45 


74473o 
744928 


3 


i5 


•41 


824893 


4-56 


175107 


i5 


46 


3- 


i5 


919762 1 


.41 


825i66 


4-56 


H4834 


14 


a 


743117 


3- 


i5 


91967711 


•41 


825439 
825 7 i3 


4-55 


1 7456i 


i3 


7453o6 


3- 


14 


919593 1 


•41 


4-55 


174287 


12 


49 


7454o4 


3- 


14 


919508! 1 


•4i 


820986 


4-55 


1 740 1 4 


11 


5o 


745683 


3- 


14 


9194241 


•41 


826259 


4-55 


173741 


10 


5i 


9-745871 


3- 


14 


9-919339 1 


• 41 


9-826532 


4-55 


10-173468 


I 


52 


746o5o 
746248 


3- 


14 


919254 1 


•41 


8268o5 


4-55 


173195 


51 


3- 


i3 


919163 1 


•41 


827078 


4-55 


172922 


7 


54 


746436 


4- 


i3 


919080 1 


•41 


827351 


4-55 


172649 


6 


55 


746624 


3- 


i3 


919000:1 
91891 5) 1 

9i883oji 


•4i 


827624 
827897 


4-55 


172376 


5 


56 


746812 


3- 


i3 


•42I 


4-54 


172103 


4 


^ 


746999 


3- 


i3 


•42 


828170 


4-54 


171830 


3 


58 


747187 


3- 


12 


9i«745!i 


•42 


828442 


4-54 


I7i558 


2 


59 


747374 


3- 


12 


918659,1 


•42| 


828715 


4-54 


171285 


I 


Co 


747562 


3-12 


9185741 


•421 


828987 
Cotang. i 


4-54 


171013 


. 




Cosine 




D. 


Sine |5G°! 


D 


Tang. 


M. 



62 


(34 


■ DEGREES.) A 


TABLE OF LOGARITHMIC 




M. 


Sine 


D. 


Cosine 


D. 


1 Tang. 


D. 


Cotang. j 





9-747562 


3-12 


9-918574 1-42 
918489 1-42 


9-828987 


4-54 


10-171013 60 


i 


747749 


3-12 


829260 


4-54 


170740 


& 


2 


7479 3 ^ 


3-12 


918404 1*42 


829532 


4-54 


170468 


3 


748123 


3. II 


9i83i8 


1-42 


829805 


4-54 


170195 


57 


4 


7483io 


3. II 


918233 


1-42 


830077 


4-54 


169923 


56 


5 


748497 


3. II 


918147 


1-42 


83o34g 


4-53 


1 6965 1 


55 


6 


748683 


3. II 


918062 


1-42 


83o62i 


4-53 


169379 


54 


I 


748870 


3. II 


917976 


1-43 


83o8 9 3 


4-53 


169107 


53 


749056 


3-io 


917891 


1-43 


83 1 1 65 


4-53 


168835 


5a 


9 


749243 


3-io 


917805 


1-43 


83i437 


4-53 


168563 


5s 


10 


749429 
9-749615 


3-io 


917719 


1-43 


83i7og 


4-53 


1 68291 


5o 


ii 


3-io 


9-917634 


1-43 


9-831981 


4-53 


io- 168019 


3 


12 


749801 


3-io 


917548 


1-43 


832253 


4-53 


167747 


i3 


749987 


3-09 


917462 


1-43 


83 2 5 2 5 


4-53 


167475 


47 


M 


750172 


3-09 


9 i 7 3 7 6 


1-43 


832796 


4-53 


167204 


46 


i5 


75o358 


3-09 


917290 


1-43 


833o68 


4-52 


166932 


45 


16 


75o543 


3-09 


917204 


i-43 


83333 9 


4-52 


1 6666 1 


44 


\l 


750729 


3-09 
3.08 


917118 


1.44 


8336n 


4-52 


i6638o 


43 


7D0914 


917032 


1-44 


833882 


4-52 


166118 


42 


*9 


751099 


3-o8 


916946 
916859 


1-44 


834 1 54 


4-52 


165846 


41 


20 


751284 


3-o8 


1-44 


834425 


4-52 


165575 


40 


21 


9-751469 


3-o8 


9-916773 


1-44 


9-834696 


4-52 


io- i653o4 


$ 


22 


75i654 


3-o8 


916687 


1-44 


834967 
835238 


4-52 


i65o33 


23 


7 5i83 9 
752023 


3-o8 


916600 


1.44 


4-52 


164762 


37 


24 


3-07 


9i65i4 


1-44 


835509 


4-52 


1 6449 1 


36 


25 


752208 


3-07 


916427 


1-44 


835 7 8o 


4-5i 


164220 


35 


26 


752392 


3-07 


9i634i 


1-44 


836o5i 


4-5i 


163949 


34 


11 


752576 


3-07 


916254 


1-44 


836322 


4-5i 


163678 


33 


732760 


3-07 


916167 


1-45 


8365 9 3 


4-5i 


163407 


32 


?9 


752944 


3-o6 


916081 


1-45 


836864 


4-5i 


!63i36 


3i 


3o 


753i28 


3-o6 


915994 


1-45 


837134 


4-5i 


162866 


3o 


3i 


9-7533i2 


3-o6 


9-915907 
915820 


1-45 


9-8374o5 


4-5i 


10-162595 


3 


32 


753495 


3-o6 


1-45 


837675 


4-5i 


162325 


33 


753619 


3-o6 


915733 


1-45 


83 79 46 


4-5i 


162054 


27 


34 


753862 


3-o5 


9 1 5646 


1-45 


8382i6 


4-5i 


161784 


26 


35 


754046 


3-o5 


915559 


1-45 


83848 7 


4-5o 


i6i5i3 


25 


36 


754229 


3-o5 


915472 


1 -45 


838 7 5 7 


4-5o 


161243 


24 


37 


754412 


3-o5 


9 1 5385 


1-45 


839027 


4-5o 


160973 


23 


38 


754595 


3-o5 


910297 


1-45 


839297 
83 9 568 


4-5o 


160703 


22 


3 9 


754778 


3-04 


91D210 


i-45 


4-5o 


i6o432 


21 


4o 


754960 


3-o4 


9i5i23 


1-46 


83 9 838 


4-5o 


160162 


20 


41 


9-755i43 


3-04 


9-9i5o35 


1-46 


9-840108 


4-5o 


io- 159892 


\l 


42 


755326 


3-o4 


9M948 


1-46 


840378 


4-5o 


159622 


43 


7555o8 


3-04 


914860 


1-46 


840647 


4-5o 


159353 


17 


44 


755690 


3-o4 


914773 


1 -46 


840917 


4-49 


1 59083 
i588i3 


16 


45 


755872 


3-o3 


914685 


1 -46 


841 187 


4-49 


i5 


46 


756o54 


3-o3 


914598 


1-46 


841457 


4.49 


158543 


14 


% 


7 56236 


3-o3 


914510 


1-46 


841726 


4-49 


158274 


i3 


756418 


3-o3 


914422 


1-46 


841996 


4-49 


1 58oo4 


12 


i 9 


7566oo 


3-o3 


9U334 


1-46 


842266 


4-49 


157734 


11 


5o 


756782 


3-02 


914246 


1-47 


842535 


4-49 


157465 


10 


5i 


9-756963 


3-02 


9-914158 


1-47 


9-842805 


4-49 


10-157195 


* 


52 


757144 


3-02 


914070 


1-47 


843074 


4-49 


156926 


53 


757326 


3-02 


913982 


1-47 


843343 


4-49 


i5665 7 
156388 


7 


54 


757507 
757688 


3-02 


913894 


1.47 


843612 


4-49 
4-48 


6 


55 


3-oi 


913806 


1-47 


843882 


i56n8 


5 1 


56 


757869 


3-oi 


913718 


1-47 


8441 5 1 


4-48 


155849 


4 1 


B 


758o5o 


3-oi 


9i363o!i-47 


844420 


4-48 


i5558o 


3 


75823o 


3-oi 


913541 1 1 -47 


844689 
844958 


4-48 


i553ii 


2 


5 9 


7584i 1 


3-oi 


9i3453 1 -47 


4-48 


i55o42 


1 1 


6o 


7585 9 i 


3-oi 


9i3365|i-47 


845227 


4-48 


i54773 


1 




Oosino 


D. 


Sine 1 55° 


Cotang. 


D. 


J?™&^. 


mJ 





BINES 


AND TANGENTb. 


(35 DEGREES. 


) 


53 


M. 


Sine 


D. 


Cosine 


D. 


Tang. 


D. 


Cotang. f 





9-758591 


3-oi 


9-9i3365 


i-47 


9-845227 


4 


48 


io- : 54773 


60 


i 


7 58 77 2 


3 


OO 


913276 




47 


845496 


4 


48 


i545o4 


5I 


2 


758 9 52 


3 


00 


913187 




48 


845764 


4 


48 


154236 


3 


759132 


3 


00 


913099 




48 


846o33 


4 


48 


153967 


57 


4 


759312 


3 


00 


9i3oio 




48 


846302 


4 


48 


i536 9 8 


56 


5 


759492 


3 


00 


912922 




48 


846570 


4 


47 


1 5343o 


55 


6 


759672 


2 


99 


912833 




43 


846839 


4 


47 


i53i6i 


54 


I 


759852 


2 


99 


912744 




48 


847107 


4 


47 


152893 


53 


76003 r 


2 


99 


912655 




48 


847376 


4 


47 


152624 


52 


9 


76021 1 


2 


99 


912566 




48 


847644 


4 


47 


i52356 


5i 


10 


760390 


2 


9 


912477 




48 


847913 


4 


47 


152087 


5o 


II 


9-760569 


2 


9-912388 




48 


9-848181 


4 


47 


io-i5i8i9 


% 


12 


760748 


2 


98 


912299 




49 


848449 


4 


47 


i5i55i 


1 3 


760927 


2 


98 


912210 




49 


848717 
848986 


4 


47 


i5i283 


47 


14 


761106 


2 


98 


912121 




49 


4 


47 


i5ioi4 


46 


i5 


761285 


2 


98 


9i2o3i 




49 


849254 


4 


47 


1 50746 


45 


16 


761464 


2 


98 


91 1942 




49 


849522 


4 


47 


150478 


44 


«7 


761642 


2 


97 


9ii853 




49 


849790 
85ood8 


4 


46 


l5o2IO 


43 


18 


761 82 1 


2 


97 


911763 




49 


4 


46 


14994a 


42 


'9 


761999 


2 


97 


91 1674 




49 


85o325 


4 


46 


149675 


4i 


20 


762177 


2 


97 


911584 




49 


85o5 9 3 


4 


46 


149407 


40 


21 


9-762306 


2 


97 


9-911495 1 


49 


9-85o86i 


4 


46 


10-149139 

14887 1 


3 9 


22 


762534 


2 


96 


911405 1 


49 


85i 129 

85i3 9 6 


4 


46 


38 


23 


7627:2 


2 


96 


91 i3i5 1 


DO 


4 


46 


143604 


37 


24 


762889 


2 


96 


911226 




DO 


85 1 664 


4 


46 


148336 


36 


25 


763067 


2 


96 


9in36 




5o 


85i 9 3i 


4 


46 


148069 


35 


26 


763245 


2 


96 


91 1046 




5o 


852199 
852466 


4 


46 


147801 


34 


27 


763422 


2 


96 


910956 
910866 




DO 


4 


46 


147534 


33 


28 


763600 


2 


9 5 




5o 


852 7 33 


4 


45 


U7267 


32 


29 


763777 


2 


9 5 


910776 




DO 


853ooi 


4 


45 


146099 


3i 


3o 


763954 


2 


9 5 


910686 1 


DO 


853268 


4 


45 


146732 


3o 


3i 


9-764131 


2 


9 5 


9-91059611 


DO 


9-853535 


4 


45 


10-146465 


2 5 


32 


764308 


2 


9 5 


9io5o6 




DO 


853802 


4 


45 


146198 


28 


33 


764485 


2 


94 


910415 




5o 


854069 


4 


45 


14593 1 


27 


34 


764662 


2 


94 


9io325 




5i 


854336 


4 


45 


145664 


26 


35 


764838 


2 


94 


910235 




5i 


8546o3 


4 


45 


145397 


25 


36 


765oi5 


2 


94 


910144 




5i 


854870 


4 


45 


i45i3o 


24 


3- 


765191 


2 


94 


91005411 


5i 


855i37 


4 


45 


144863 


23 


38 


760367 


2 


94 


909963 1 


5i 


855404 


4 


45 


144596 


22 


39 


765544 


2 


9 3 


909873 




Dl 


855671 


4 


44 


144329 


21 


4o 


765720 


2 


9 3 


909782 




5i 


855 9 38 


4 


44 


J 44062 


20 


4i 


9.765896 


2 


9 3 


9-909691 




DI 


9-856204 


4 


44 


10-143796 


19 


42 


766072 


2 


9 3 


909601 




DI 


856471 


4 


44 


U352o 18 


43 


766247 


2 


9 3 


909510 1 


DI 


856737 


4 


44 


143263 


H 


44 


766423 


2 


9 3 


909419 1 


DI 


857004 


4 


44 


142996 


16 


45 


766598 


2 


92 


90932811 


52 


857270 


4 


44 


142730 


i5 


46 


766774 


2 


92 


909237 1 


D2 


857537 


4 


44 


142463 


14 


a 


766949 


2 


92 


909146,1 


D2 


85 7 8o3 


4 


44 


142197 


i3 


767124 


2 


92 


9ooo55 1 


52 


858069 


4 


44 


141931 


12 


49 


767300 


2 


92 


908964 1 


52 


858336 


4 


44 


141664 


11 


5o 


767475 


2 


9i 


908873 1 


D2 


858602 


4 


43 


141398I 10 
io-i4n32 9 


5i 


q- 767649 


2 


9 1 


9-908781 1 


52 


9-858868 


4 


43 


5> 


767824 


2 


91 


908690 1 


D2 


859134 


4 


43 


140866 8 


53 


767999 
768173 


2 


91 


908399' 1 


D2 


859400 


4 


43 


140600 


I 


54 


2 


91 


9085071 1 


52 


859666 
8D9932 


4 


43 


Uo334 


55 


768348 


2 


90 


908416! 1 


53 


4 


43 


140068 


5 


56 


768522 


2 


90 


908324 1 


53 


860198 


4 


43 


139802 


4 


n 


768697 


2 


9 c 


9oS233;i 


53 


860464 


4 


43 


139536 3 


768871 


2 


90 


908 1 41 ' 1 


53 


860730 


4 


43 


139270! 2 


59 


769045 


2 


90 


908049 1 


53 


86099$ 


4 


43 


1 39005 1 
138739 


Co 


76921? 

Cosine 


2 90 


00795811 


53 


86i2Si 
Cotaog 


4-43 




~ 


D. 


Sine - 154° 


T 


X 


Tan*. 


,2«kj 



54 


(30 


DEGREES.) A 


TABLE OF LOGARITHMIC 




"MT 


Sine 


D. 


Cosine | D. 


Tang. 


D. 


Cotaag 


60 


o 


9-769219 


2-90 


9-9079581-53 9-861261 


4-43 


10-138739 
138473 


i 


7 6 9 3 9 3 


2 


^ 9 


907866, 1 


-53! 86i52 7 


4 


43 


3 


2 


769366 


2 


89 


907774:1 
907682' 1 


53 1 861792 
53 j 862058 


4 


42 


138208 


3 


769740 


2 


89 


4 


42 


137942 


n 


4 


769913 


2 


89 


907590-1 


53 


862323 


4 


42 


137677 


5 


770087 


2 


& 


907498 1 


53 


86258 9 


4 


42 


137411 


55 


6 


770260 


2 


90740611 


53 


862854 


4 


42 


137146 


54 


1 7 


■770433 


2 


88 


907314 Ii 


54 


863iiq 

863385 


4 


42 


i3688i 


53 


6 


770606 


2 


88 


907222 1 


54 


4 


42 


i366i5 


52 


9 


770779 


2 


88 


907129 1 


54 


86365o 


4 


42 


i3635o 


5i 


10 


770952 


2 


88 


907037 




54 


8639 1 5 


4 


42 


i36o85 


5o 


ii 


9-771125 


2 


88 


9-906945 




54 


9-864180 


4 


42 


io-i3582o 


% 


F2 


771298 


2 


87 


906852 




54 


S64445 


4 


42 


135555 


13 


771470 


2 


87 


906760 




54 864710 


4 


42 


135290 


% 


U 


771643 


2 


87 


906667 




54 


864975 


4 


41 


i35o25 


i5 


771815 


2 


87 


906575 




54 


865240 


4 


41 


134760 


45 


16 


771987 


2 


87 


906482 




54 


8655o5 


4 


41 


134495 
i3423o 


44 


\l 


772109 


2 


S 


906389 




55 


865770 


4 


4i 


43 


77233i 


2 


86 


906296 




55 


866o35 


4 


41 


i33g65 


42 


19 


7725o3 


2 


86 


906204 




55 


8663oo 


4 


4i 


133700 


41 


20 


772675 


2 


86 


9061 1 1 




55 


866564 


4 


4i 


133436 


40 


21 


9-772847 
773018 


2 


86 


9-906018 




55 


9-866829 


4 


4i 


10-133171 


ll 


22 


2 


86 


9o5o25 




55 


867094 


4 


4i 


132906 


23 


773190 


2 


86 


9o5832 




55 


86 7 358 


4 


4i 


132642 


37 


24 


77336i 


2 


85 


905739 
905645 




55 


867623 


4 


4i 


1 323 77 


36 


25 


773533 


2 


85 




55 


867887 


4 


4i 


i3?n3 


35 


26 


773704 


2 


85 


9o5552 




55 


8681 52 


4 


4o 


1 3x848 


34 


27 


7738 7 5 


2 


85 


905459 




55 


868416 


4 


4o 


i3i584 


33 


28 


774046 


2 


85 


9o5366 




56 


868680 


4 


4o 


i3i32o 


32 


I 9 


774217 


2 


85 


905272 




56 


868945 


4 


4o 


i3io55 


3i 


3o 


774388 


2 


84 


9o5i7Q 

9-9o5o85 




56 


869 2 00 


4 


4o 


1 30794 


3o 


3i 


9-774558 


2 


84 




56 


9-869473 


4 


4o 


io-i3o527- 


3' 


32 


774729 


2 


84 


904992 




56 


869737 


4 


40 


i3o263 


33 


774899 


2 


84 


904898 




56 


870001 


4 


4o 


1 29999 
129735 


27 


34 


775070 


2 


84 


904804 




56 


870265 


4 


40 


26 


35 


775240 


2 


84 


9047 1 1 
904617 




56 


870529 


4 


40 


129471 


25 


36 


77 5 4io 


2 


83 




56 


870793 
871057 


4 


40 


129207 


24 


ll 


77558o 


2 


83 


904523 




56 


4 


40 


128943 


23 


775750 


2 


83 


904429 
904335 




57 


8 7 i32i 


4 


40 


128679 
128415 


22 


09 


775920 


2 


83 




57 


8 7 i585 


4 


4o 


21 


4o 


7760QO 


2 


83 


904241 




57 


871849 


4 


3 9 


I28i5i 


20 


4' 


9-776259 


2 


83 


9-904147 




57 


9-872112 


4 


3 9 


10-127888 


\l 


42 


776420 
776598 


2 


82 


904053 




5 7 


872376 


4 


3 9 


127624 


43 


2 


82 


903959 




57 


872640 


4 


3 9 


1 27360 


17 


44 


776768 


2 


82 


903864 




57 


872903 


4 


3 9 


127097 


16 


45 


776937 
777106 


2 


82 


903770 




57 


873167 


4 


3 9 


126833 


i5 


46 


2 


82 


903676 




57 


873430 


4 


3 9 


126570 


14 


% 


777275 


2 


81 


9o358i 




57 


873694 
873967 


4 


39 


i263o6 


i3 


777444 


2 


81 


903487 




57 


4 


3 9 


1 26043 


12 


49 


777613 


2 


81 


903392 




58 


874220 


4 


39 


125780 


n 


5o 


777781 


2 


81 


903298 




58 


874484 


4 


39 


I255i6 


10 


5i 


9 777950 


2 


81 


9«9o32o3 




58 


9-874747 


4 


39 


IO-I25253 


i 


52 


778119 


2 


81 


903 1 08 




58 


875010 


4 


3 


1 24990 


53 


778287 


2 


80 


9o3oi4 




58 


875273 


4 


124727 


I 


54 


778455 


2 


80 


902919 
902824 




58 


8 7 5536 


4 


38 


1 24464 


55 


778624 


2 


80 




58 


875800 


4 


38 


124200 


5 


56 


778792 


2 


80 


902729 




58 


876063 


4 


38 


123937 


4 


S 


778960 


2 


80 


902634 




58 


876326 


4 


38 


123674 


3 


779128 


2 


80 


902539 




5 9 


876589 


4 


38 


1 234i 1 


2 


59 


779295 


2 


79 


902444 




5 9 


8 7 685 1 


4 


38 


123149 


1 


60 


779463 


2.79 


902349 




5 9 


877114 


4-38 


122886 







Cosine 


D. 


Sine 


53°l Cotang. 


D. 


Tang. 



blNKS AND TANGENTS. (37 DEGREES.) 



55 



6 


Sine 


1 D ' 


Cosine i 


D. 


Tang. 


i D - 


Cotang. 


60 


9-779463 


a 


•79 


9-902349' 


i-5 9 


9-877114 


4-38 


10-122886 




77 9 63i 


2 


•79 


902253 


i-5 9 


877377 


4-38 


122623 


tl 


2 


779798 


2 


"9 


902158 1 


i-5 9 


877640 


4-38 


122360 


3 


779966 


2 


79 


902063 


i-5 9 


877903 


4-38 


122097 


5 7 


4 


78oi33 


2 


•79 


901967 


i-5 9 


8 7 8i65 


4-38 


I2i835 


56 


5 


78o3oo 


2 


78 


901872! 


i-5 9 


878428 


4-38 


121672 


55 


6 


780467 


2 


78 


901776, 


i.5 9 


878691 


4-38 


121309 54 


7 


780634 


2 


78 


901681J 


i-5 9 


878933 


4-3 7 


121047 


53 


8 


780801 


2 


78 


90 1 585; 


i.5 9 


879216 


4-37 


120784 


52 


9 


780968 


2 


78 


901490 


i-5 9 


879478 


4-3 7 


120522 


5i 


10 


781 i34 


2 


•78 


901394 


i-6o 


879741 


4-37 


I 202D9 


DC 


ii 


9«78i3oi 


2 


•77 


9-901298 


i-6o 


9-88ooo3 


4-37 


IO-II9997 


3 


12 


781468 


2 


•77 


901202; 


[•60 


880265 


4-3 7 


1 19735 


i3 


i8/634 


2 


77 


90II06| 


i- 60 


880528 


4-37 


I I9472 


% 


14 


781800 


2 


77 


9OIOIO: 


i- 60 


880790 


4-3 7 


IIG2I0 


i5 


781966 


2 


77 


900914' 

90081 8 | 


i-6o 


881002 


4-3 7 


1 1 8948 


45 


16 


782132 


2 


77 


i-6o 


88i3i4 


*-3 7 


1 1 8686 


44 


\l 


782298 


2 


76 


9OO722! 


[•60 


881576 


4-3 7 


118424 


43 


782464 


2 


76 


900626 


i-6o 


881839 


4-3 7 


118161 


42 


19 


782630 


2 


76 


900529 
900430 


[•60 


882101 


4.37 


1 17899 
1 17637 


4i 


20 


782796 


2 


76 


[•61 


882363 


4-36 


4o 


21 


9-782961 


2 


76 


9-900337 


[-61 


9-882625 


4-36 


10-117375 


ii 


22 


783i2 7 


2 


76 


900240 


•61 


882887 
883i48 


4-36 


1 171 i3 


23 


783292 
783458 


2 


75 


900144 


[•61 


4-36 


1 1 6852 


37 


24 


2 


75 


900047 


•61 


8834io 


4-36 


1 16590 


36 


25 


783623 


2 


75 


899951 
899854 


•61 


883672 


4-36 


1 16328 


35 


26 


7 83 7 88 


2 


75 


-6i 


883 9 34 


4-36 


1 16066 


34 


11 


783 9 53 


2 


75 


899757 


•61 


884196 
884457 


4-36 


n58o4 


33 


7841 18 


2 


75 


899660 


•61 


4-36 


1 1 5543 


32 


29 


784282 


2 


74 


899564 


•61 


884719 


4-36 


u528i 


3i 


3o 


784447 


2 


74 


899467 


-62 


884980 


4-36 


Il5o20 


3o 


3i 


9-784612 


2 


74 


9-899370 


-62 


9-885242 


4-36 


10-114758 


It 


32 


784776 


2 


74 


899273 


•62 


8855o3 


4-36 


1 14497 


33 


784941 


2 


74 


899176! 


•62 


885 7 65 


4-36 


1 14235 


27 


34 


7°5 1 o5 


2 


74 


899078 

898981 


•62 


886026 


4-36 


1 13974 


26 


35 


785269 


2 


73 


•62 


886288 


4-36 


113712 


25 


36 


785433 


2 


73 


898884 


-62 


886549 


4-35 


ii345i 


24 


u 


785597 


2 


73 


898787 


•62 


886810 


4-35 


113190 


23 


785761 


2 


73 


898689 


-62 


887072 


4-35 


1 1 2928 


22 


39 


785925 


2 


73 


898592 


•62 


887333 


4-35 


1 1 2667 


21 


40 


786089 
9-786252 


2 


73 


898494 


•63 


887594 


4-35 


1 1 2406 


20 


41 


2 


72 


9-898397 


•63 


9.887855 
8881 16 


4-35 


10-112145 


% 


42 


786416 


2 


72 


898299 


•63 


4-35 


1 1 1884 


43 


786579 


2 


72 


898202 


•63 


888377 


4.35 


111623 


\l 


44 


786742 


2 


72 


898104 


•63, 


88863 9 


4-35 


1 1 i36i 


45 


786906 


2 


72 


898006 


•63' 


888900 


4-35 


IIIIOO 


i5 


46 


787069 


2 


72 


897908 


-63 


889160 


4-35 


1 10840 


14 


47 


787232 


2 


7i 


897810 


•63 


889421 


4-35 


1 1 0579 


i3 


48 


787395 
7 8 7 55 7 


2 


7' 


897712 


•63 1 


889682 


4-35 


iio3iS 


12 


49 


2 


7i 


897614 


•63i 


889943 


4-35 


1 1 0057 


11 


5o 


787720 


2 


7i 


897516 


•63! 


890204 


4-34 


109796 


10 


5i 


9.787883 


2 


7i 


9-897418! 


•64! 


9-890465 


4-34 


10-109535 


9 


52 


788045 


2 


7* 


897320 


•64 


890725 


4-34 


109275 8 


53 


788208 


2 


7i 


897222 


•64 


890986 


4.34 


1090141 7 


54 


788370 


2 


70 


897123 


•64! 


891247 


4-34 


108753 




55 


7 88532 


2 


70 


897025 


•64; 


891507 


4-34 


io84o3 
1082J2 


5 


56 


788694 
788856 


2 


70 


896926 


•64 


891768 


4-34 


4 


57 


2 


70 


896828 ] 


•64! 


892028 


4-34 


10-1972 


3 


58 


789018 


2 


70 


896729 


•64 


892289 


4-34 


107711 


2 


5 9 


789 1 80 


2 


70 


896631!] 


•64 


892549 


4-34 


1 0745 1 


1 


60 


789342 


2-69 


896532 1 


•64! 


892810 


4-34 


107190 





D. 


Siue N 


Cotang. i 


D. 


Tiiiiar. 


M. 



56 


(38 DEGREES.) A 


TABLE OF LOGARITHMIC 




to. 


Sine 


D. 


Cosine 


D. 


_Tang. 


D. 


Cctang. 


* 





9-789342 


2-69 


9-896532 


1-64 


9-892810 


4-34 


10-107190 

1 06930 


60 


I 


789504 


2-69 


896433 


1-65 


893070 


4-34 


u 


2 


789665 


2-69 


896335 


1-65 


8 9 333 1 


4-34 


1 06669 


3 


789827 


2-69 


896286 


1-65 


893591 
8 9 385i 


4-34 


106409J 57 


4 


7S9988 


2-69 


896137 


1-65 


4-34 


1 06 1 49 56 


5 


79° '49 


2-69 
2-68 


8 9 6o38 


1-65 


8941 1 1 


4-34 


io588g 


55 


6 


7903 10 


8 9 5o3 9 
895840 


1-65 


894371 


4-34 


io562o 
io5368 


54 


I 


790471 


2-68 


1-65 


894632 


4-33 


53 


790632 


2-68 


895741 


1-65 


894892 
895152 


4-33 


io5io8 


52 


9 


790793 
790954 


2-68 


895641 


1-65 


4-33 


104848 


5i 


10 


2-68 


895542 


1-65 


895412 


4-33 


104588 


5o 


ii 


9-79iii5 


2-68 


9-895443 


1-66 


9-895672 


4-33 


10-104328 


% 


12 


791275 


2-67 


8o5343 


1-66 


895932 


4-33 


104068 


i3 


79U36 


2-67 


895244 


i-66 


896192 
896402 


4-33 


io38o8 


47 


U 


791396 
791757 


2-67 


895145 


1-66 


4-33 


io3548 


46 


i5 


2-67 


895045 


1-66 


8967 1 2 


4-33 


io3288 


45 


16 


791917 


2-67 


894045 


1-66 


80697 1 


4-33 


103029 


44 


3 


792077 


2-67 


894846 


1-66 


897231 


4-33 


102769 


43 


792237 


2-66 


894746 


1-66 


897491 
897751 


4-33 


102509 


42 


19 


792397 


2-66 


894646 


1-66 


4-33 


102249 


4i 


20 


792537 


2-66 


894546 


1-66 


898010 


4-33 


101990 
10-101730 


40 


21 


9.792716 


2-66 


9-894446 


1-67 


9-898270 


4-33 


ll 


22 


792876 


2-66 


894346 


1-67 


8 9 853o 


4-33 


101470 


23 


793o35 


2-66 


894246 


1-67 


898789 


4-33 


101211 


ll 


24 


7 9 3 1 o5 
793354 


2-65 


894146 


1-67 


899049 
899308 


4-32 


1 0095 1 


36 


25 


2-65 


894046 


1.67 


4-32 


100692 


35 


26 


79^5i4 


2-65 


893946 


1-67 


899568 


4-32 


1 0043 2 


34 


3 


793673 


2-65 


893846 


1-67 


899827 


4-32 


1 00 1 73 


33 


793832 


2-65 


893745 


1-67 


900086 


4-32 


099914 


32 


29 


793991 


2-65 


893645 


1.67 


900346 


4-32 


099654 


3i 


3o 


79400 


2-64 


893544 


1-67 


900605 


4-32 


099395 

10-099136 

098876 


3o 


3i 


9- 7943o8 


2-64 


9 -8 9 3444 


i-68 


9.900864 


4-32 


It 


32 


794467 


2-64 


893343 


1.68 


901124 


4-32 


33 


794626 


2-64 


893243 


1-68 


90i383 


4-32 


098617 


ll 


34 


794784 


2-64 


893142 


i-68 


901642 


4-32 


098358 


35 


794942 


2-64 


893041 


1-68 


901901 


4-32 


098099 


25 


36 


790101 


V64 


892940 
892839 


i-68 


902160 


4-32 


097840 


24 


37 


795259 


2-63 


1-68 


902419 


4-32 


097581 


23 


38 


790417 


2-63 


892739 


i-68 


902679 
902938 


4-32 


097321 


22 


3 9 


79 55 7 5 


2-63 


892638 


i-68 


4-32 


097062 


21 


4o 


795733 


2-63 


8 9 2536 


i-68 


903197 


4-3i 


096803 


20 


4 1 


9.795891 


2-63 


9-892435 


1.69 


9-9o3455 


4-3i 


10-096545 


■9 


42 


796049 


2-63 


892334 


1-69 


903714 


4-3i 


096286J 18 


43 


796206 


2-63 


892233 


1-69 


903973 


4-3i- 


096027 


\l 


44 


796364 


2-62 


892132 


1-69 


904232 


4-3i 


090768 


45 


796521 


2-62 


892030 


1.69 


904491 


4-3i 


095509 


i5 


46 


796679 


2-62 


891929 


1-69 


904700 


4-3-1 


095250 


14 


% 


7 9 6836 


2-62 


891827 


1.69 


905008 


4-3i 


094992 


i3 


796993 


2-62 


891726 


1.69 


905267 


4-3i 


094733 


12 


49 


797i5o 


2-6l 


891624 


1.69 


9o5526 


4-3i 


094474 


11 


5o 


797307 


2-6l 


891523 


1-70 


905784 


4-3i 


094216 


10 


5i 


9-797464 


2- 6l 


9-891421 


1.70 


9-906043 


4-3-1 


10-093957 


§ 


52 


797621 


2-6l 


891319 


1-70 


906302 


4-3i 


093698 


53 


797777 


2-6l 


891217 


1.70 


9o656o 


4-3i 


093440 


7 


54 


797934 


2-6l 


891115 


1-70 


9068 10 


4-3i 


093181 


6 


55 


798091 


2- 6l 


891013 


1-70 


907077 


4-3i 


092923 


5 


56 


798247 


2-6l 


89091 1 | 
890809' 


/•70 


907336 


4 3i 


092664 


4 


u 


798403 


2-6o 


1-70 


907594 
907852 


4-3i 


092406 


3 


798560 


2-60 


890707! 


1-70 


4-3i 


092148 


2 


59 


798716 


2-60 


890605 


1.70 


9081 11 


4-3o 


091889 


60 


798872 


2-6o 


890503! 


1.70 


908369 


4-3o 


091 63 1 







Cosine 


D. 


Sine |tfl°! 


Cotang. 


D. 


~Tang7~| 


m7 





SINES 


AND TANGENTS. 


(39 DEGREES." 


) 


5*3 


M. 
o 


Sine 


D. 


Cosine j D. 


Tang. 


D. 


Cotang. ! 


9-798872 


2-60 


9 -89o5o3! 1 -70 


9-908360 


4-3o 


10-091631 60 


i 


799028 


2 


60 


890400(1 


7i 


908628 


4 


3o 


091372 5o 
091 1 14 58 


a 


799184 


2 


60 


890298 1 


71 


908886 


4 


3o 


3 


799339 


2 


5 9 


890195 1 


7i 


909144 


4 


3o 


090856 j 57 


4 


799495 


2 


5 9 


89009-3 ; 1 


7i 


909402 


4 


3o 


090598 56 


5 


799601 


2 


5 9 


809990' I 


7i 


909660 


4 


3o 


090340 


55 


6 


799806 


2 


5 9 


8898881 


7i 


909918 


4 


3o 


090082 


54 


7 


799962 


2 


5 9 


889785' I 


7i 


910177 


4 


3o 


089823 


53 


8 


8001 17 


2 


5 9 


889682 ' I 


71 


910435 


4 


3o 


089565 


52 


9 


800272 


2 


58 


889579)1 


71 


910693 
910931 


4 


3o 


089307 | 5i 


10 


800427 


2 


58 


889477,1 


7i 


4 


3o 


080049 


5o 


u 


9-8oo582 


2 


58 


9-889374 I 




9-911209 


4 


3o 


10-088791 
o88533 


% 


I 12 


800737 


2 


58 


889271 I 


72 


91 146- 


4 


3o 


1 l3 


800892 


2 


58 


88gi68ii 


72 


911724 


4 


3o 


088276 


47 


U 


801047 


2 


58 


889064' 1 


72 


911982 


4 


3o 


088018 


46 


i5 


801201 


2 


58 


88S96 1 ; I 


"2 


912240 


4 


3o 


087760 


45 


16 


80 1 356 


2 


5 7 


888858 i 


^2 


912498 


4 


3o 


087502 


44 


\l 


8oi5ii 


2 


5 7 


8887551 1 


72 


912756 


4 


3o 


087244 


43 


80 1 665 


2 


57 


88865i 1 1 


72 


9i3oi4 


4 


29 


086086 


42 


»9 


801819 
801973 


2 


57 


888548|i 


72 


913271 


4 


29 


086729 


41 


20 


2 


57 


888444 ' 1 


"3 


913529 


4 


29 


086471 


4o 


21 


9-802128 


2 


57 


9-88834i| 1 


73 


9-9*3787 


4 


29 


10-086213 


ll 


22 


802282 


2 


56 


888237 1 


73 


914044 


4 


29 


085956 


23 


802436 


2 


56 


888i34 1 


-3 


9U3o2 


4 


29 


085698 


37 


24 


802589 
802743 


2 


56 


888o3o|i 


73 


9U56o 


4 


29 


085440 


36 


25 


2 


56 


887926! 1 


73 


9U817 


4 


29 


o85i83 


35 


26 


802897 


2 


56 


8878221 


-3 


915075 


4 


29 


084925 


34 


27 


8o3o5o 


2 


56 


88771811 


-3 


915332 


4 


29 


084668 


33 


28 


8o32o4 


2 


56 


887614 1 


73 


915590 


4 


29 


084410 


32 


& 


8o3357 


2 


55 


887510 1 


73 


9i5847 


4 


29 


084 1 53 


3i 


3o 


8o35u 


2 


55 


887406 1 


74 


916104 


4 


29 


083896 
io-o83638 


3o 


3i 


9-8o3664 


2 


55 


9-887302 1 


"4 


9-916362 


4 


29 


ll 


32 


8o38i 7 


2 


55 


887198!] 


74 


916619 


4 


29 


o8338 1 


33 


803970 


2 


55 


887093 1 1 
8869^9 1 
886885! 1 


74 


916877 


4 


29 


o83i23 


27 


34 


804123 


2 


55 


74 


9Hi34 


4 


29 


082866 


26 


35 


804276 


2 


34 


74 


917391 


4 


29 


082609 


25 


36 


804428 


2 


54 


886780:1 


-4 


917648 


4 


29 


o82352 


24 


h 


8o458i 


2 


54 


886676 1 


-4 


917905 


4 


3 


082095 


23 


38 


804734 


2 


54 


8865 7 i|i 


74 


9i8i63 


4 


081837 


22 


3 9 


804886 


2 


54 


886466 1 


U 


918420 


4 


28 


o8i58o 


21 


4o 


8o5o39 


2 


54 


886362 1 


7 5 


018677 


4 


28 


o8i323 


20 


4! 


9- 800191 


2 


54 


9.886257 1 


V 


9-918934 


4 


28 


1 • 08 1 066 


19 


42 


8o5343 


2 


53 


886i5 2 !i 


73 


919191 


4 


28 


080809 


18 


43 


8o5495 


2 


53 


886047! 1 


75 


919448 


4 


28 


o8o552 


17 


44 


8o5647 


2 


53 


885942 ji 
88583 7 1 


75 


919705 


4 


28 


080295 
o8oo38 


16 


45 


800799 
8059D1 


2 


53 


75 


919962 


4 


28 


i5 


46 


2 


53 


885 7 32 




75 


920219 


4 


28 


079781 


14 


47 


806 1 o3 


2 


53 


885627 




75 


920476 


4 


28 


079524 


i3 


48 


806254 


2 


53 


885522 




75 


920733 


4 


28 


079267 


12 


49 


806406 


2 


52 


885416 1 


13 


920990 


4 


28 


0190101 11 


5o 


806557 


2 


52 


8853iiji 


76 


921247 


4 


28 


078753 


10 


5i 


9-806709 


2 


52 


9-8852o5'i 


76 


9-92i5o3 


4 


28 


10-078497 


I 


>2 


806860 


2 


52 


885iooi 


-6 


921760 


4 


28 


078240 


53 


80701 1 


2 


52 


884Q94 1 


-6 


Q220T7 


4 


28 


077983 


7 


54 


807163 


2 


52 


8848891 


-6 


922274 


4 


28 


077726 


6 


55 


807314 


2 


52 


8847S3 1 


76 


Q22530 


4 


28 


077470 


5 


56 


807465 


2 


5i 


884677 1 


-6 


922187 


4 


28 


077213 


4 


a 


807615 


2 


5i 


884572 1 


"6 


923044 


4 


28 


076956 


3 


807766 


2 


5i 


884466 1 


76 


923300 


4 


28 


076700 


2 


5, 


807917 


2 


5i 


88436o| 1 


76 


923557 


4 


27 


076443 


1 


* 


808067 


2-5l 


884204 1 


77 


9 238i3 


4 


27 


076187 
Tang. 



_M. 


Coeine 


D. 


Sine !oO c 


Cotang. 


D. 



5b 


(4( 


J DEGREES.) A 


TABLE OF LOGARITHMIC 




M. 


Sine 


D. 


Cosine 


D. | Tang. 


D. 


Cotang, 




o 


9 • 808067 
808218 


2-5l 


9-884254 


1-77J 9-923813 


4-27 


10-07618-] 


60 


i 


2-5l 


884148 


1 -77 


92407c 


4-27 


07593c 


sii 


2 


8o8368 


2-5l 


884042 


x -77 


92432- 


4-27 


075673 


3 


8o85i 9 


2-5o 


883 9 36 


1.77 


924583 


4-27 


075417 


5 7 

56 


4 


8o366 9 


2-5o 


883829 


1.77 


92484c 


4-27 


07516c 


5 


808819 


2-5o 


883723 


1.77 


925096 


4-27 


074904 


55 


6 


808969 


2-5o 


883617 


*-77 


9253D2 


1 4-27 


074648 


54 


7 


809119 


2-5o 


8835io 


1 -77 


925600 
925865 


4-2 7 


074301 


53 


8 


809269 


2-5o 


883404 


1.77 


4-2 7 


074135 


52 


9 


809419 


2-49 


883297 


1.78 


926122 


4-2 7 


073878 


5i 


IC 


809669 

9-809718 


2-49 


883191 


1.78 


926378 


4-27 


073622 5o 


i j 


2-49 


9-883o84 


1-78, 9-926634 


4-2 7 


I C -073366 49 


12 


809868 


2-49 


882077 
882871 


1 -78: 926890 


1 4-2 7 


07311c 


48 


i3 


810017 


2-49 


1-78 927147 


; 4-2 7 


072853 


47 


14 


81 01 67 


2-49 
2.48 


882764 


1.78 


927403 


4-2 7 


072597 


46 


i5 


8io3t6 


882657 


1.78 


927609 
927910 


4-27 


072341 


45 


16 


8io465 


2.48 


88255o 


i. 7 » 


4-27 


072085 


44 


17 


810614 


2-48 


882443 


1.78 


928171 


4-27 


071829 


43 


18 


810763 


2-48 


882336 


1-79 928427 


4-27 


071573 


42 


•9 


8 1 09 1 2 


2-48 


882229 


1.79 


928683 


4-2 7 


071317 


41 


20 


8 r 1 06 1 


2-48 


882121 


1.79 


928940 


4-2 7 


071060 


40 - 


21 


9-811210 


2-48 


9-882014 


I.79 


9-929196 
929452 


4-2 7 


10 070804 


u 


22 


8u358 


2-47 


881907 


1.79 


4-2 7 


070548 


23 


81 1D07 


2-47 


88 1 799 


1.79 


929708 


4-2 7 


070292 


37 


24 


8n655 


2-47 


881692 


1.79 


929964 


4-26 


070036 


36 


25 


81 1804 


2-47 


88 1 584 


1.79 


930220 


4-26 


069780 


35 


26 


81 1952 


2-47 


881477 


1.79 


93o475 


4-26 


069525 


34 


27 


81 2 1 00 


2-47 


88i36 9 


1.79 


930731 


4-26 


069269 
069013 


33 


28 


812248 


2-47 
2-46 


881261 


i-8o 


930987 


4-26 


32 


2 9 


8i23g6 


881 1 53 


i-8o 


93i243 


4-26 


068757 


3i 


3o 


812044 


2-46 


881046 


i-8o 


931499 


4-26 


o685oi 


3o 


3i 


9-812602 


2-46 


9-88o 9 38 


1. 80 


9-931755 


4-26 


10-068245 


29 


32 


812840 


2-46 


88o83o 


1. 80 


932010 


4-26 


067990 


28 


33 


812988 


2.46 


880722 


i-8o 


932266 


4-26 


067734 


27 


34 


8i3i35 


2-46 


88o6i3 


1. 80 


932522 


4-26 


067478 


26 


35 


8i3283 


2-46 


88o5o5 


1.80 


932778 


4-26 


067222 


25 


36 


8i343o 


2-45 


8S0397 


1. 80 


933o33 


4-26 


066967 


24 


3 7 


8i35 7 8 


2-45 


880289 


1. 81 


933289 


4-26 


0667 ' ' 


23 


38 


8i3725 


2-45 


880180 


1. 81 


933545 


4-26 


o66455 


22 


3 9 


8i38 7 2 


2-45 


880072 


1. 81 


9338oo 


4-26 


066200 


21 


4o 


814019 


2-45 


879963 


1. 81 


934o56 


4-26 


065944 


20 


4i 


9-814166 


2-45 


9-879855 


1. 81 


9-9343ii 


4-26 


1 • 065689 


I? 


42 


8i43i3 


2-45 


£79746 


1.81 


934567 


4-26 


065433 


43 


814460 


2-44 


879637 


1. 81 


934823 


4-26 


065177 


17 


44 


814607 


2-44 


879529 


1. 81 


935078 


4-26 


064922 


16 


45 


8i4753 


2-44 


879420 


i-8i 


935333 


4-26 


064667 


i5 


46 


814900 


2-44 


8793 n 


i-8i 


935589 


4-26 


06441 1 


14 


47 


81 5o46 


2-44 


879202 


1-82 


935844 


4-26 


064 1 56 


i3 


48 


SiSigS 


2-44 


8790931 


1-82 


936 1. .10 


4-26 


063900 


12 


49 


8i5339 


2-44 


878984I 


1-82 


936355 


4-26 


063645 


11 


5o 


8 1 5485 


2-43 


878875 

9-878766 


1.82 


936610 


4-26 


063390 
io-o63i34 


10 


5i 


9-8i563i 


2-43 


1-82 


9-936866 


4-25 


3 


52 


815778 


2-43 


8 7 8656 


1.82 


937121 


4-25 


062879 


53 


815924 


2-43 


8785471 


1-82 


937376 


4-25 


062624 


7 


54 


816069 


2-43 


878438 


1.82 


937632 


4-25 


062368, ( 


55 


8i62i5 


2-43 


878328 


[•82 


937887 


4-25 


0621 i3 


5 


56 


8i636i 


2-43 


878219 


[-83 


938142 


4-25 


06 1 858 


4 


^7 


816007 


2-42 


878109 


[-83 


938398 
9-38653 


4-25 


061602 


3 


58 


816602 


2-42 


877999 


-83 


4-25 


061347 


2 


5 9 


8167981 


2-42 


877800 


[-83 


9 38 9 o8 


4-25 


061092 
060837 
Tnv.g. 


1 


6o 


816943 

Cosine | 


2-42 


8777S0 


[-83 


939163 


4-25 





I). 


Sine 


49° 


Cotansr. 


~~i)7~ 





BiJNES AND TANGENTO. 


(41 DEGREES. 


) 


59 


p£" 


Sine 


D. 


Cosine | D. 


Tang. 


D. 


Cotang. 




c 


g. 816943 


2-42 


9.877780 i-83 
8776701-83 


9939163 


4-25 


i 0-060837 


60 


i 


817088 


2 


•42 


939418 


4-25 


o6o582 


U 


2 


817233 


2 


•42 


877560! i-83 


939673 


4-25 


060327 


3 


817379 


2 


.42 


877450' 1 -83 


939928 


4-25 


060072 


57 


4 


817524 


2 


•41 


877340 i-83 


94oi83 


4-25 


059817 


55 


5 


817668 


2 


•41 


877230 1.84 


940438 


4-25 


359562 


55 


6 


817813 


2 


•41 


877120 1-84 


940694 


4-25 


059306 


54 


I 


817958 


2 


•41 


877010 1-84 


940949 


4-25 


03905 1 
058796 
058542 


53 


8i8io3 


2 


•41 


876890I1.84 
876789I1.84 


941204 


4-25 


52 


9 


818247 


2 


•41 


941458 


4-25 


5i 


10 


8i83o2 
o-8i8536 


2 


•41 


87667s 1 1- 84 


9417U 


4-25 


058286 


5o 


ii 


2 


.40 


g. 876568 I.84 


9-941968 


4-25 


io-o58o32 


% 


13 


818681 


2 


.40 


876457i I- 84 


942223 


4-25 


057777 


i3 


818825 


2 


.40 


876347 j I- 84 


942478 


4-25 


057522 


47 


14 


818969 
8191 i3 


2 


.40 


8 7 6236| 1. 85 


942733 


4-25 


057267 


46 


; 5 


2 


40 


876125,1-85 


942988 


4-25 


057012 


45 


16 


819257 


2 


40 


876014 i-85 


943243 


4-25 


056757 


44 


17 


819401 


2 


.40 


875904 i i-85 


943498 


4-25 


o565o2 


43 


18 


819545 


2 


39 


875793!l85 
875682! 1. 85 


943752 


4-25 


056248 


42 


19 


819689 


2 


39 


944007 


4-25 


055993 


4i 


20 


819832 


2 


39 


875571(1-85 


944262 


4-25 


055738 


4o 


21 


9.819976 


2 


39 


9-87545911.85 
87534811.85 


9.944517 


4-25 


10-055483 


3 9 


22 


820120 


2 


39 


944771 


4-24 


055229 


38 


23 


820263 


2 


3 9 


875237 i-85 


945026 


4-24 


054974 


37 


24 


820406 


2 


a 


875126 i-86 


945281 


4-24 


0547 1 9 


36 


25 


82o55o 


2 


875oi4|i-86 


945535 


4-24 


054465 


35 


26 


820693 
82o836 


2 


38 


874903 i-86 


945790 


4-24 


054210 


34 


3 


2 


38 


874791 1 1 -86 


946045 


4-24 


053955 


33 


820979 


2 


38 


874680! i-86 


946299 
946554 


4-24 


053701 


32 


29 


821122 


2 


38 


8745681 1- 86 


4-24 


053446 


3i 


3o 


82126D 


2 


38 


874456 i-86 


946808 


4-24 


o53ig2 


3o 


3i 


9-821407 


2 


38 


9.87/34411-86 


9-947063 


4-24 


10.052937 


2 


32 


82i55o 


2 


38 


874232 ji -87 


9473i8 


4-24 


052682 


33 


821693 


2 


37 


874i2iji-87 


947572 


4-24 


052428 


27 


34 


82i835 


2 


37 


874009 1-87 


947826 


4-24 


032174 


26 


35 


821977 


2 


? 7 


87389611.87 


948081 


4-24 


051919 


25 


36 


822120 


2 


37 


873784! 1-87 


948336 


4-24 


031664 


24 


u 


822262 


2 


37 


873672J1.87 


948590 


4-24 


o5i4io 


23 


822404 


2 


37 


873560 1-87 


948844 


4-24 


o5u56 


22 


39 


822546 


2 


37 


873448 1 1. 87 


949099 


4-24 


030901 


21 


4o 


822688 


2 


36 


873335 1.87 


949353 


4-24 


050647 


20 


4i 


9-822830 


2 


36 


9-87322311 .87 


9 949607 


4-24 


io-o5o393 


\l 


42 


822972 


2 


36 


8731 10 i-88 


949862 


4-24 


o5oi38 


43 


823i 14 


2 


36 


872998 
872885 


1.88 


950116 


4-24 


049884 


17 


44 


823255 


2 


36 


i-88 


950370 


4-24 


049630 


16 


45 


823397 


2 


36 


872772 i-88 


9D0625 


4-24 


049375 


i5 


46 


823539 


2 


36 


87265911.88 


950879 


4-24 


049121 
048867 


14 


8 


82368o 


2 


35 


872547 


1-88 


95u33 


4-24 


i3 


823821 


2 


35 


872434 


1-88 


9 5i388 


4-24 


048612 


12 


4 9 


823963 


2 


35 


872321 


1-88 


951642 


4-24 


048358 


11 


5o 


824104 


2 


35 


872208 


1.88 


951896 


4-24 


048104 


10 


5i 


9-824245 


3 


35 


9-872095 1-89 
8719811-89 
87 1 8681 1. 89 


9-952i5o 


4-24 


10-047850 


I 


52 


824386 


2- 


35 


9524o5 


4-24 


o475 9 5 


53 


824527 


2- 


35 


952659 
95291J 


4-24 


047341 


I 


54 


824668 


2- 


34 


871755 1. 80 


4-24 


047087 
046833 


55 


824808 


2- 


34 


871641 


1-89 


953167 


4-23 


5 


56 


824949 


2- 


34 


871528 


1-89 


953421 


4-23 


046579 
046325 


4 


u 


825090 
82523o 


2- 


34 


871414 


1-89 


953675 


4-23 


3 


2- 


34 


871301 


1.89 


953929 
954i83 


4-23 


04607 1 


2 


5 9 


825371 


2- 


34 


871187 


i-8o 


4-23 


045817 


1 


6o 


8255n 


2-34 


871073:1.96 


954437 


4-23 


045563 





Cosine 


D. 


Sine |48q 


Cotang. J 


' D. ! 


Tang. 


&L 



60 


(42 


DEGREES.) A TABLE OF LOGARITHMIC 




~M.~ 


Sine 


D. 


Cosine | D. 


Tang. 


D. 


Cotang. 







9-8255u 


2-34 


9-871073,1 -90 


9-954437 


4-23 


io-o45563 


60 


i 


82565i 


2 


33 


870960 1 


•90 


954691 


4-23 


045309 


n 


2 


82^791 


2 


33 


870846 1 


.90 


954945 


4-23 


o45o55 


3 


825g3i 


2 


33 


870732 1 


•90 


955200 


4-23 


044800 


57 


4 


80607 1 


2 


33 


870618,1 


• 9 c 


955454 


4-23 


044546 


56 


5 


82621 1 


2 


33 


870504 1 


■90 


. 9 55 707 


4-23 


044293 


55 


6 


82635i 


2 


33 


870390 1 


•90 


955961 


4-23 


044039 


54 


I 


826491 


2 


33 


8702761 


•90 


9562i5 


4-23 


043785 


53 


826631 


2 


33 


870161 ; 1 


•90 


956469 


4-23 


o43 53 1 


52 


9 


826770 


2 


32 


870047J1 


•91 


956723 


4-23 


043277 


5i 


10 


826910 


2 


32 


869933 1 


•91 


956977 


4-23 


o43o23 


5o 


ii 


9-827049 


2 


32 


9.869818; 1 


•91 


9-957231 


4-23 


10.042769 


% 


12 


827189 
827328 


2 


32 


869704!! 


.91 


957485 


4-23 


o425iS 


i3 


2 


32 


86 9 589 1 


.91 


9 5 77 39 
957993 


4-23 


042261 


47 


14 


827467 


2 


32 


869474I1 


.91 


4-23 


042007 


46 


i5 


827606 


2 


32 


869360' 1 


•9' 


958246 


4-23 


o4h54 


45 


16 


827745 


2 


32 


86 9 245ii 


.91 


9585oo 


4-23 


041 5oo 


44 


n 


827884 


2 


3i 


86gi3o!i 


.91 


9 58 7 54 


4-23 


041246 


43 


18 


828023 


2 


3i 


86901 5 1 


.92 


959008 


4-23 


040992 
040738 


42 


19 


828 162 


2 


3i 


86890011 


•92 


959262 


4-23 


4i 


20 


8283oi 


2 


3i 


868785] 1 


•92 


959516 


4-23 


040484 


4o 


21 


9.828439 


2 


3i 


9-86867011 


•92 


9 -959769 


4-23 


10- 04023 I 


ll 


22 


828578 


2 


3i 


8685551 1 


•92 


960023 


4-23 


039977 


23 


828716 


2 


3i 


868440 1 


•92 


960277 


4-23 


o39723' 37 


24 


828855 


2 


3o 


868324 1 


•92 


96o53 i 


4-23 


039469 36 


2D 


828993 
8291 3i 


2 


3o 


868209 1 
86809J 1 


•92 


060784 


4-23 


039216 


35 


26 


2 


3o 


•92 


961038 


4-23 


o38 9 62 


34 


3 


829269 


2 


3o 


867978 1 
867862 1 


•93 


961 291 


4-23 


038709 


33 


829407 


2 


3o 


. 9 3 


961545 


4-23 


o38455 


32 


29 


829545 


2 


3o 


867747 1 


•93 


961799 
962052 


4-23 


o382oi 


3i 


3o 


829683 


2 


3o 


86 7 63 1 1 


.93 


4-23 


037948 


3o 


3i 


9-829821 


2 


29 


9- 867*51 5 1 


■ 9 3 


9-962306 


4-23 


10-037694 


3 


32 


829959 


2 


29 


867399 1 

867283 1 


- 9 3 


962560 


4-23 


037440 


33 


830097 


2 


29 


•93 


962813 


4-23 


037187 


27 


34 


830234 


2 


29 


867167 1 


• 9 3 


963067 


4-23 


036933 26 


35 


83o372 


2 


29 


867051 1 


■93 


963320 


4-23 


o3668o 


25 


36 


83o5o9 


2 


2 9 


866935 1 


•94 


963574 


4-23 


o36426 


24 


37 


83o646 


2 


29 


86681911 
866703 1 


■94 


963827 


4-23 


o36i73 


23 


38 


830784 


2 


8 


■94 


964081 


4-23 


035919 
o35665 


22 


39 


830921 


2 


866586 1 


.94 


964335 


4-23 


21 


40 


83io58 


2 


28 


866470 1 


•94 


964588 


4-22 


o354i2 


20 


41 


9 -83 1 195 
83i33 3 


2 


28 


9-866353 1 


•94 


9 • 964842 


4-22 


io-o35i58 


\l 


42 


2 


28 


866237 1 


•94 


965095 


4-22 


o349o5 


43 


83i46g 


2 


28 


866120 1 


•94 


965349 


4-22 


o3465i 


17 


44 


83 1 606 


2 


28 


866004 1 


• 9 5 


965602 


4-22 


034398 


16 


45 


831742 


2 


28 


865887 1 


•95 


965855 


4-22 


o34U5 


i5 


46 


831879 


2 


28 


865770 1 


• 9 5 


966105 


4-22 


033891 


14 


47 


832oi5 


2 


27 


865653! 1 


■ 9 5 


966362 


4-22 


033638 


i3 


48 


832i52 


2 


27 


865536ji 
865419 1 


■ 9 5 


966616 


4-22 


033384 


12 


49 


832288 


2 


27 


• 9 5 


966869 


4-22 


o33i3i 


11 


5o 


832425 


2 


27 


8653o?ji 


• 9 5 


967123 


4-22 


032877 


10 


5i 


9-83256i 


2 


2 7 


9-865i85 1 


• 9 5 


9-967376 


4-22 


10 032624 





52 


8326 97 


2 


27 


865o68 1 


• 9 5 


967629 


4-22 


032371 


8 


53 


832833 


2 


11 


864950 1 
864833 1 


•9D 


967883 
968136 


4-22 


032117 


7 


54 


832969 


2 


.96 


4-22 


o3i864 


6 


55 


833io5 


2 


26 


8647 1 6 1 1 


.96 


968389 


4-22 


o3i6u 


5 


56 


833241 


2 


26 


8645981 


.96 


968643 


4-22 


o3i357 


4 


n 


8333 77 


2 


26 


864481:1 


.96 


968896 


4-22 


o3no4 


3 


8335i2 


2 


26 


864363 j 1 


•96! 


969149 


4-22 


o3o85i 


2 


59 


833648 


2- 


26 


864245 1 


•96! 


969403 


4-22 


030597 


1 


60 


833783 


2- 


26 


864127 1 


.96 


969656' 


4-22 


o3o344 





(losme 


P. 


Sine 14?°' 


Cotang. 1 


D. 


Tarur. 


mT 





5INE8 AND TANGENTS. 


(43 DEGREES. 


) 


61 


M. 


Sine 


K 


Cosine D. 


Tang. 


D. 


Cotanor. 







9-833783 


2-26 


9-864127 1-96 


9-969656 


4-22 


io-o3o344 


~6o~ 


i 


833qi9 


2 


•25 


864010 1 


.96 


969909 


4 


•22 


030091J 5o 
02 9 838| 58 


2 


834054 


2 


•25 


8638 9 2 1 


•97 


970162 


4 


22 


3 


834i8 9 
834325 


2 


•25 


863774 1 


•97 


970416 


4 


22 


029584 


57 


4 


2 


2C 


863656 1 


•97 


970669 


4 


22 


029331 


56 


5 


83446o 


2 


25 


863538|i 


•97 


970922 


4 


22 


029078 


55 


6 


834595 
834730 


2 


25 


863419 1 


•97 


971175 


4 


• 22 


028825 


54 


7 


2 


25 


8633oi 1 


•97 


97U29 


4 


22 


028571 


53 


8 


834865 


2 


25 


863 1 83 1 


•97 


971682 


4 


22 


0283i8 


52 


9 


834999 


2 


24 


863o64 1 


■97 


971935 


4 


22 


028065 


5i 


10 


835i34 


2 


24 


862946 1 


.98 


972188 


4 


22 


027812 5o 


ii 


9.835269 
8354o3 


2 


24 


9-862827 1 


.98 
.98 


9-972441 


4 


22 


10-027559 49 


12 


2 


24 


862709 1 


972694 


4 


22 


027306 


48 


i3 


835538 


2 


24 


8623901 


.98 


972948 


4 


22 


027052 


47 


U 


835672 


2 


24 


862471 1 


.98 


973201 


4 


22 


026799 


46 


i5 


835807 


2 


24 


862353 1 


.98 


973454 


4 


22 


026546 


45 


16 


835g4i 


2 


24 


862234' 1 


.98 


973707 


4 


22 


026293 


44 


n 


836o 7 5 


2 


23 


862II3I 


.98 


973960 


4 


22 


026040 


43 


18 


836209 
836343 


2 


23 


861996 I 


.98 


9742i3 


4 


22 


025787 


42 


19 


2 


23 


861877:1 


.98 


974466 


4 


22 


025534 


41 


20 


836477 


2 


23 


86i 7 58 1 


•99 


974719 


4 


22 


025281 


40 


21 


9. 8366 n 


2 


23 


9-86i638 1 


•99 


9-974973 


4 


22 


10.025027 


3 9 


22 


836 7 45 


2 


23 


86i5i 9 1 


■99 


975226 


4 


22 


024774; 38 


23 


806878 


2 


23 


861400 1 


•99 


975479 


4 


22 


024521 37 


24 


837012 


2 


22 


861280 1 


•99 


975732 


4 


22 


0242681 36 


25 


837146 


2 


22 


861161:1 


•99 


975985 


4 


22 


024015 


35 


26 


837279 


3 


22 


861041 1 


•99 


976238 


4 


22 


023762 


34 


27 


837412 


2 


22 


860922 ' 1 

860802 1 


■99 


976491 


4 


22 


023509 


33 


28 


83 7 546 


2 


22 


•99 


976744 


4 


22 


023256 


32 


29 


837679 


2 


22 


860682 2 


•00 


976997 


4 


22 


o23oo3 


3i 


3o 


837812 


2 


22 


86o562 2 


•00 


97725o 


4 


22 


022750 


3o 


3i 


9-837945 


2 


22 


9-860442 2 


•00 


9-9775o3 


4 


22 


10-022497 29 
022244 28 


32 


838o 7 8 


2 


21 


86o322 2 


•00 


977756 


4 


22 


33 


8382 1 1 


2 


21 


860202 2 


•00 


978009 


4 


22 


021991 27 


34 


838344 


2 


21 


860082*2 


•00 


978262 


4 


22 


021738 26 


35 


- 838477 


2 


21 


859962 2 


■00 


9785i5 


4 


22 


021485 25 


36 


8386io 


2 


21 


859842^2 


•00 


978768 


4 


22 


021232 24 


37 


838742 


2 


21 


859721 2 


•0, 


979021 


4 


22 


020979] 23 


38 


838875 


2 


21 


859601 2 


•01 


979274 


4 


22 


020726! 22 


3 9 


839007 


2 


21 


859480 2 


•01 


979527 


4 


22 


020473; 21 


40 


839140 


2 


20 


859360 2 


■01 


979780 


4 


22 


020220 20 


4i 


9-839272 


2 


20 


9-859239^ 


•01 


9-93oo33 


4 


22 


10-019967; 19 
OI97U l8 


4? 


83 9 4o4 


2 


20 


85gi 192 


•01 


980286 


4 


22 


43 


83 9 536 


2 


20 


858998 <? 


•01 


98o538 


4 


22 


OI9462 17 


44 


83 9 668 
83 9 %o 


2 


20 


8583 77 !2 


•01 


980791 


4 


21 


0I020-)! 10 


45 


2 


20 


858756'2 


•02 


981044 


4 


21 


018956! l5 


46 


83 99 32 


2 


20 


858635 2 


•02 


981297 


4 


21 


018703, 14 


% 


840064 


2 


T 9 


8585i4'2 


•02 


93i55o 


4 


21 


oi845o i3 


840196 


2 


19 


8583 9 3 2 


•02 


981803 


4 


21 


018197 12 1 


49 


84032S 


2 


'9 


858272,2 


•02 


982036 


4 


21 


017044' 11 


5o 


840459 


2 


19 


858i5i 2 


■02 


982309 


4 


21 


017691! 10 

10-017438 9 

oi7i86 : 8 


5i 


9.840591 


2 


'9 


9 •358029' 2 
85 79 o3 ( 2 


•02 


9-982562 


4 


21 


52 


840722 


2 


19 


■02 


982814 


4 


21 


53 


84o854 


2 


19 


85 77 86 : 2 


■02 


983067 


4 


21 


016933 7 


54 


840985 


2 


;g 


857665 2 


•o3 


983320 


4 


21 


016680 6 


55 


841 1 16 


2 


857543 2 


•o3 


933573 


4 


21 


016427J 5 


56 


841247 


2 


18 


857422 2 


■ o3 


983826 


4 


21 


016174 4 


u 


841378 


2 


18 


8573oo!2 


• 0] 


984079 


4- 


21 


015921 3 


84:509 


2 


18 


857178:2 


•o3 


98433i, 


4- 


21 


015669 2 
oi54i6| I 


59 


841640 


2 


18 


85 7 o56i2 


•o3 


984584' 


4- 


21 


6o 


_84i77i 
Cosine 


2-l8 


856934 1 2 


■o3 


984837 


4- 


21 


oifi63 o_ 




D. 


Sic i Us ! 


Cotftng. 1 


D. 


Tang. 1 M. 



62 


(44 DEGREES.) A TABLE OP LOGARITHMIC 




M. 


Sine 


1 D. 


Cosine 


D. I Tang. 


D. 


Cotang. 
io-oi5i63 


60 





9-841771 


I 2-18 


9-856 9 34 
8568i2 


2-o3 9-984837 


4-21 


i 


841902 


2-l8 


J-o3 


985090 


4-21 


014910 


It 


2 


842033 


1 2-l8 


856690 


>-o4 


985343 


4-21 


014657 


3 


842i63 


' 2-17 


856568 


>-o4 


985596 


4-21 


014404 


u 


4 


842294 


: 2-17 


856446 


>-o4 


983848 


4-21 


014152 


5 


842424 


; 2-17 


856323 


>-o4 


986101 


4-21 


013899 


55 


6 


842555 


; 2-17 


856201 • 


> • 04 


986354 


4-21 


013646 


54 


2 


842685 


I 2-17 


856078 


>-o4 


986607 


4-21 


013393 


53 


8428i5 


i 2.17 


855o56 ' 
855833 


>-o4 


986860 


4-21 


oi3i4o 


52 


9 


842946 


2.17 


^•04 


987 1 1 2 


4-21 


012888 


5i 


10 


843076 


! 2-17 


855 7 u 


>-o5 


98 7 365 


4-21 


012635 


5o 


u 


9-843206 


2-l6 


9-855588 


>-o5 


9-987618 


4-21 


10-012382 


8 


12 


843336 


2-l6 


855465 ' 


>-o5 


987871 


4-21 


012129 


i3 


843466 


2- l6 


855?/ t2 ' 


>-o5 


988123 


4-21 


01 1877 


47 


U 


8435g5 


2-l6 


855219 ' 


>-o5 


988376 


4-21 


01 1624 


46 


i5 


843725 


2-l6 


855096 : 


>-o5 


988629 


4-21 


011371 


45 


16 


843855 


2-l6 


854973 : 
85485o ' 


j.o5 


988882 


4-21 


011118 


44 


13 


843984 


2-l6 


•o5 


989134 


4-21 


010866 


43 


844H4 


2-l5 


854727 : 


• 06 


989387 


4-21 


oio6i3 


42 


«9 


844243 


2-l5 


8546o3 5 


• 06 


989640 


4-21 


oio36o 


4i 


20 


844372 


2-l5 


854480 5 


• 06 


989893 


4-21 


010107 


4o 


21 


9-844502 


2-l5 


9-854356: 


• 06 


9-990145 


4-21 


10-009855 


ll 


22 


84463i 


2-l5 


854233 : 


• 06 


990398 
990631 


4-21 


009602 


23 


844760 


2-l5 


854109 : 


• 06 


4-21 


009349 


ll 


24 


844889 
845oi8 


2-l5 


853 9 86 2 
853862 : 


• 06 


990903 


4-21 


009097 


25 


2-l5 


■ 06 


991 1 56 


4-21 


008844 


35 


26 


845 1 47 


2-l5 


853 7 38 : 


• 06 


991409 


4-21 


008591 


34 


27 


845276 


2-14 


8536i4 2 


•07 


991662 


4-21 


oo8338 


33 


28 


8454o5 


2-14 


853490 : 


■07 


991914 


4-21 


008086 


32 


?° 


845533 


2-14 


853366 : 


•07 


992167 


4-21 


007833 


3i 


3o 


845662 


2-14 


853242 2 


• 07 


992420 


4-21 


007580 


3o 


3i 


9-845790 


2-14 


9-853n8 2 


■ 07 


9-992672 


4-21 


10-007328 


ll 


32 


845919 


2-14 


852994 2 
85286o 2 


•07 992925 


4-21 


007075 


33 


846047 


2-14 


• 07 


993178 


4-21 


006822 


27 


34 


846175 


2-14 


852745 : 


•07 


99343o 


4-21 


006570 


26 


35 


8463o4 


2-14 


852620 2 


• 07 


993683 


4-21 


006317 


25 


36 


846432 


2-l3 


852496 2 


• 08 


993936 


4-21 


006064 


24 


ll 


84656o 


2-l3 


852371 2 


.08 


994189 


4-21 


oo58n 


23 


846688 


2-l3 


852247 2 


• 08 


994441 


4-21 


oo5559 


22 


3 9 


846816 


2-l3 


852122 2 


.08 


994694 


4-21 


oo53oo 


21 


4o 


846944 


2-l3 


851997 2 
9-851872 2 


• 08 


994947 


4-21 


oo5o53 


20 


4i 


9-847071 


2-l3 


• 08 


9-995199 
995452 


4-21 


10-004801 


a 


42 


847199 


2-l3 


85i 7 47 2 


• 08 


4-21 


004548 


43 


847327 


2-l3 


85i622 2 


• 08 


995705 


4-21 


004295 


3 


44 


847454 


2-12 


85 1 497 2 


• 09 


99 5 9 57 


4-21 


004043 


45 


847582 


2- 12 


85i372 2 


• 09 


996210 


4-21 


003790 
oo3537 
oo3285 


i5 


46 


8477 9 
847836 


212 


85 1 246 2 


• 09 


996463 


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